I'll post a few gameplay tips here and edit the post as I think of more. Last edited: 4/17/2015
Undo and Redo: The beta client has the ability to undo and redo moves; this is done by left-clicking the left or right arrows at the right side of the top information bar.
Race Choice: I recommend that you play first as one of the two Ashdar factions. The other factions have some missing functionality, and holes in their ship selection. In particular, Humans start without a viable homeworld, and can be difficult to play your first time out.
Fleet selection: when you left-click on a fleet in the strategic map, you are presented with the fleet pane at the bottom of the screen. In this pane, a left-click on a ship brings toggles its selection, and a right-click inverts the selection state of the entire fleet. Clicking on the "?" icon brings up the ship status screen.
Note that this feature works in every fleet pane, including before a battle (if you are the attacker, you can choose to leave some ships -- such as colony ships or transports -- out of the battle), and when examining the trade pool.
Transports: Your empire starts with an empty Transport. Transports can be used to ferry population, or to establish trading routes to provide income. Transports can move population or ground troops manually by clicking on the Embark/Disembark icons (either from the fleet pane, the ship status display, or a colonist/tank icon in the planet screen), or they can be sent to the Trade Pool.
Transports can be assigned to the Trade Pool, from which they will automatically a) transport food between colonies when needed, b) transport population from one colony to another when instructed to, or c) generate income from trade routes. You can hover your mouse pointer over the Trade Pool icon at the top of the screen (yellow rocket icon) to view a summary of your Trade Pool's activities. Wherever you see this yellow rocket icon, there is a function linked to the Trade Pool.
You can send transports to the Trade Pool by clicking on the "Start Trading" rollover icon on the ship in the fleet pane, or by enabling the "Assign New Transports to Trade" option in the Options screen to automatically send newly constructed transports directly to the Trade Pool. Transports take 3 turns after they are sent to the pool before they are ready to start trading or transporting food or colonists. You can view the transports in the pool by clicking in the Trade Pool icon at the top of the main screen. From this screen, you can recall transports from the pool by selecting the transports you want to remove from the pool, clicking "Recall to Star", and selecting a destination. Transports sent to the Trade Pool won't start generating income until you have a second colony to trade with, but at least being in the trade pool will mean you don't have to pay upkeep every turn for it.
Surplus Food will automatically be transported to colonies that need it at a rate of 1 Food per Cargo space. Transports start with 5 Cargo spaces each, and this is increased through technologies. Cargo space that is not being used to transport Food or colonists will be used to generate income through trading. The amount of income generated will depend on the number and type of trade routes available, which improves with technologies and certain diplomatic agreements (like Sign Trade Charter).
Moving Population: Transports in the pool can be used to move population, if they are not already transporting food. If you left-click on the "Move Population" mouseover icon on a planet in the bottom pane, or shift-left-click on a planet or on a population icon in almost any screen (including the Planetary Report), this will activate the population move UI (select a destination star and planet, and if there are available transports in the pool, they will stop trading and perform the population transfer). If there is more than one type of colonist on the planet, you will be shown a dialogue in which you can choose how many of what type of colonist you want to move. After completing the transfer the transports will automatically return to the trade pool (after a short delay). The transport will not generate income or be available for more population moves until it has returned to the pool.
You can also move population with a shift-left-click on a planet or on a population icon in almost any screen (including the Planetary Report), this will activate the population move UI (select a destination star and planet, and if there are available transports in the pool, they will stop trading and perform the population transfer). After completing the transfer the transports will automatically return to the trade pool (after a short delay). The transport will not generate income or be available for more population moves until it has returned to the pool.
Different types of population have different biome requirements, and so may be better suited to certain types of planets than others. Many planets have a variety of biomes, and so having multiple different types of colonists that can best exploit each biome (such as a land-dwelling species and a sea-dwelling species for an Earthlike "Garden" world with both oceans and land) will increase the maximum population cap. If a colonist type will increase the population cap, you will see a green modifier below the population icon while hovering over the target planet in the population move UI.
Early Exploration: Many star systems are infested with pirates and marauders, so explore carefully. I recommend that you either scout a system before sending in your colony ship, or escort the colony ship with the scout. The scout is unarmed, but it may draw fire off the colony ship and give it a chance to retreat. Scouts and other military vessels can easily retreat from enemies, but civilians vessels like colony ships and transports require an extra turn before they can retreat, and so will be vulnerable to enemy fire.
Setting Production: To set planet production, left-click on a planet (either on the "?" icon or directly elsewhere on the planet) to enter the Planet Information screen. The square pane at lower left shows the planet's current production, and how many turns are left. Right now, the game always auto-selects something to build, so be sure to check when a planet has finished production to make sure it's producing what you want it to be producing. Change production by left-clicking on the production pane, and then left-clicking on a choice from the "Choose Production" menu. If you have enough currency, you will have the option to buy the item immediately for the listed cost. Otherwise, the item will be produced in the listed number of turns. You may change production any time; currently, the amount of resources you've already spent are carried over to the new production target. There is not currently a production queue, though this is a planned addition. [This UI is due for an overhaul.]
Designing Ships: Before you build a new class, check the Ship Designer to make sure it has the components that you need (make sure Scouts have Warp Lane Amplifiers, Troop Transports have Planetary Assault Forces, etc.). Enter the Ship Designer by left-clicking on an item in the production menu, or by choosing "Ship Designer" from the game menu. In the Ship Designer, fill slots on the left with available components on the right.
Tactical Combat: In tactical combat, left-click to select a ship, and right-click on a valid move target (indicated by a dimmed image of the ship) or right-click on an enemy ship in range to fire.
A ship will attempt to fire all of its weapons of each type with each click in this order: fighters, missiles, offensive beam weapons, and defensive point-defense weapons. If you would like to have it fire only some of its weapons (such as when a missile base wants to spread its missiles across multiple small targets instead of overkilling just one), left-click or right-click on the weapon description in the bottom center pane to increment or decrement the number of available weapons that will fire. Note that point-defense weapons will not normally fire at another ship unless you issue a second attack command; they are normally reserved for use against fighters and missiles, and can fire during the opponent's turn.
You can also shift-left-drag to band-box select multiple ships, and shift-click to add and remove ships from the selection group.
Research: In the Research screen (accessed by left-clicking on the blue test tube icon at the top information bar), you can browse the available technologies by left-clicking on items in the list or on hyperlinks in the tech description pane, but to actually change the technology being researched, you must left-click on the checkbox at the left of the name of the desired technology, or click on the "Set Research" button at the lower left. Left-clicking on the field icon to the left of each technology in the list will bring up a list of all of the technologies in that field. This can be useful in identifying what prerequisites you need for a technology that isn't yet listed in the center list.
Upgrading and Refitting Ships: In the production menu, you can select an existing design, or you can Refit an existing ship that's in a fleet in the same system, or choose New Design which will create a new ship design class. If you wish to change an existing class (instead of creating a new one), you can shift-left-click on the class (or you can exit the production menu, hit Escape to bring up the game menu, and choose "Ship Design"). Note that changing a class design doesn't affect already existing ships of that class; each existing ship must be manually refitted. Changing a ship design will change all the ships of that class that are still under construction but haven't yet been completed.
Capturing an Enemy Planet: Once you have defeated a planet's defenses, you have two options to directly attack the planet itself. First, you can bombard the population with your military vessels, killing them one unit at a time. If you kill all the population, the colony will be destroyed. Currently, bombardment does not destroy infrastructure, only population.
Second, if your fleet includes Troop Transports (or any ship equipped with Planetary Assault Forces), you can invade and capture the population. Each Planetary Assault Force unit can subdue 1 unit of population; you must have enough shuttles to subdue the entire population to be able to invade. If the enemy planet has more population than you have invading shuttles, you can destroy the surplus population one unit at a time with bombardment, and then invade and capture the rest.
Those races with a working Troop Transport need the Docking Fields tech. This requires the prerequisite techs Small Craft, Orbital Construction, Tachyon Physics, and Deflector Screens. Those races that do not currently have a working Troop Transport (Yoral) must research the subsequent Carrier technology and build a custom assault ship that includes planetary assault forces.
Note: there is a recent change that allows civilian Transports to carry Planetary Assault Forces as well.
Planetary Report: Left-clicking on the orange wrench icon at the top left of the strategic map will activate the Planetary Report pane, which will show you the current production status of all of your planets. You can leave this pane open all the time if you find it useful. Left-clicking on an item brings up the Planet Information Screen for that planet, and you can even move population from the Planetary Report by shift-left-clicking on an item.
Diplomacy: Not all of the diplomacy features are working in the current build, so you may find yourself at war with multiple factions at a time. Plan accordingly.
Ground Bases: The design of your ground bases can be changed by entering the Ship Designer from the game menu (shift-clicking in the production pane won't work) and editing one of the three "Planet Defenses" items. Any changes are immediately reflected in existing bases; there's no need to "refit" them.
Playing as Humanity: The Human player has a different start than the other factions; they don't have a homeworld and instead start with an outpost around a gas giant and a small fleet of transports and colony ships. You will have very limited income until you can find a suitable planet and establish your first colony. Before you disembark the population on your transports to your new colony, make sure that you have enough food production, otherwise they will starve. It may make sense to rush-buy a Farm if you want to get started quickly.