Just to remix things a bit, as others have said, concentrate early research on getting long range frigates. These often form the very backbone of a fleet, and most definitely do so during the earlier parts of a game(perhaps ending the game outright if it is a small 1v1). Don't bother much with other research early on, as getting more ships instead provides a bigger benefit. In general the good early research includes: long range frigates with flak frigates after that to protect them, arctic colonizion and trade ports on the civilian side (the Advent can throw up a culture building early too).
In terms of capital ships, just stick with the one freebie early on. Expand your fleet supply one tick and get more frigates for some quick firepower. As for starter capital ships, if you're doing small 1v1 against the AI: TEC can't go wrong with a Sova Carrier for early combat, Advent go well with a Progenitor Mothership and a Halcyon carrier to follow a bit later, and the Vasari can do well with the Evacuator colonizer or a Devastator battleship and either can be followed by a Skirantra carrier. The TEC are the easiest faction to use as you're learning the game though. They have a ton of cheap ships coupled with a good economy.
For starbases, don't try rushing starbase tech early on. If you're on a decently sized map, say with 8 or more gravity wells per player, you can plop one down at a chokepoint. But don't leave it alone. Put a frigate factory there along with some repair structures. Starbases are quite weak until they get 1-2 levels in defense. But that costs a fair bit of money. Your fleet should be the ones to focus on early game defense. A single repair structure can make all the difference for a defending fleet.
Relating to that, don't waste a bunch of resources on static defenses like turrets. If you need an early game choke point, just setting down a frigate factory with one repair structure can be enough. If you know the enemy has a decent fleet in the area, consider putting a single turret next to the frigate factory and repair structure. That adds some good punch to frigates as they defend an area.
When it comes to actual expansion, grabbing your asteroid next door and a single planet nearby is enough to get you started pumping out fleet. From there look for chokepoints to secure, or neutral extractors you want to claim. Neutral extractors can by very important to early economic booms, as they provide a far faster income rate than standard extractors at your colonies. From there you can send a second capital around to neutral worlds within your secured sector to colonize places and get some free experience.
For race specific generic opening strategies to get you started:
TEC - LRM frigates followed by repair structures and the Hoshiko Robotics cruiser, then some Garda flak frigates as needed, on the civilian side go for colonization and trade ports. TEC thrives on credits, and their cheap ships which repair on the go thanks to the Hoshiko lets them leverage a strong economy even as they fight. You should ideally have credits falling out of your pockets even as you spend them, purchasing extra resources when needed to continue your empire's growth.
Advent - takes some getting used to compared to the TEC. They need more research to get their Illuminators and trade ports. Luckily the Disciples are cheap and your freebie capital ship(whether Progenitor or Halcyon) fights well alongside them. Getting culture up early can help balance out their economy, especially since they'll want to colonize an extra world or two early on if safely possible. As Darvin3 said in another thread, often times you'll stop at 3 civilian research labs and focus on military from there on. Guardians pair up quite well with the Progenitor, and their tier 4 Repulsion ability totally changes the battlefield in favor of the Illuminators.
Vasari - focused on quick mobile expansion/combat. Assailants are great if you can find even a single neutral crystal extractor, and are generally worth selecting over the somewhat maligned Skirmisher. Sentinels can be tossed in later as needed. Phase Missiles are used on so many Vasari ships, strikecraft and structures that it makes it really easy to pick up at least the tier 1 research relatively early. Combine it with strong capital ship abilities like Nano-disassemblers on the Evactuator(which makes your early fleet specialized capital ship hunters, especially once it hits level 4 and you can pick up Gravity Warhead to prevent escapes by damaged enemy capitals), or Scramble Bombers on the Skirantra(which is about to be toned down to a far more reasonable level) and you have an early game advantage despite fielding fewer ships. Keep the pressure up though, since your credit economy will take forever to ramp up. The Vasari focus more on holding extractors and trading in their extra resources for credits, either to other players or the black market. Lastly, their starbase tech is only tier 2, and is deployed from the otherwise humble Vasari colonizing frigate. This plays directly into their quick expansion philosophy, allowing you to effectively seal off neutral extractors for yourself(preferably by seizing a chokepoint ahead of them).
