Which ship is the best to begin the game with?
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You can beat the game about as reliably with with any ship, so the differentiator for "best ship" is going to be entirely based on score. Score is influenced by how many beacons you visit, how many ships you destroy, and how much scrap you get. Scrap isn't influenced by ship, so that can be ignored. The stealth ship may be able to visit more beacons since you can more reliably navigate nebulas out the gate, which will slow down the rebel fleet. But that only applies in a rare situation, and it pays for it by having to run away sometimes until you can find shields.
So what you want is a ship that can blow up everything.
For this, I'd recommend:
For similar reasons, I would recommend against the engi and zoltan ships. The engi ship can easily find itself in a position where it can't do much damage (I ran into an early Defense II drone and neither side could damage the other), and the zoltan ship's strength is being able to wait out damage rather than dish it out.
That's a somewhat subjective question. Still, the following ship shave strong starts IMO:
Despite the lack of cloaking the Fed Cruiser is super strong overall.
I would probably rank those to ships the best of what I have used.
Zoltan - never used the Type B. But the Type A gets a Halberd beam which will immediately cut through any sector 1 and most sector 2 shields. While bad for boarding, Zoltan are otherwise exceptional crew member supplying free power. The Zoltan shield is excellent and will mitigate quite a bit of of damage. I don't think this is quite as good as the Fed, as it needs a little more to come into it's own.
Slug Cruiser (no experience with the type B). It does have dual lasers and a Bio Beam which are really strong (and you can use both off the right bat). It also has doors 2. Still, I would rank it just behind the best ships due to only 2 starting crew.
I don't have experience with the stealth or Crystal ship. I'm new to the Bulwark and don't feel I can adequately comment there.