I wouldn't worry much about the Geonosians at first. Their ships are good, but they as characters aren't, and since ships need their pilots to be geared up to be any good, the first ships you should focus on should be ones whose pilots are also good.
That's why you start with Biggs, Vader, the Imperial Tie Fighter, and Boba Fett's Slave-I as your top priorities. Biggs is the best tank ship in the game bar none, and the other three are all strong attackers who provide plenty of target locks for Biggs, and all four pilots have their uses - Biggs alongside Wedge and other rebels, Vader in Empire and Sith teams, the Tie Pilot in Empire teams and certain raid teams, and Fett with Bounty Hunters or just as a decent stand-alone addition to teams until your roster is more diverse.
Other good ships to grab earlier include Maul's Scimitar, Unmasked Kylo's Tie Silencer, and the two Phoenix ships (though be warned that the Phoenix squad is very hard to gear past G8, and since each ship has 3 pilots it's a lot of work to strengthen those two).
You might use some lesser ships early on that you'll not want to stick with later just because you have the pilot, too. Wedge is a good example of this - you'll want him to go with Biggs on normal teams, but his ship is just kind of okay. It hits hard, but has nothing else to recommend it. But that's still better than gearing up a pilot who's weak just for his ship while your roster is small. Other examples of this might be the Jedi Consular's Fighter (you'll have him anyway since you started with him, good-ish ship), Rex's ARC (great character, easy to gear, decent ship), and Ahsoka's Jedi Fighter (very easily accessible character you might have, not hard to gear, relatively good ship).
I wouldn't recommend the Resistance X-Wing. It's not bad per se, but Resistance Pilot is difficult to gear and not that good even once she's geared, and the X-Wing itself isn't great either.
Longer-term, the First Order Tie Fighter is great, but it's hard to get the pilot farmed quickly, so I wouldn't jump on him too early.
As far as your capital ship goes, Tarkin is probably the best one to use first. It's debatable whether his ship or Ackbar's Home One is the best of the starting three, but Tarkin is undoubtedly the best pilot of the three. Ackbar has his uses, but not as many as Tarkin.