My first group went with some of the homebrew ships I statted, but after more books came out, they had goals that were sadly not reached.
My previous group LOVED the Ghtroc as a starting ship. The pilot immediately took ranks in Skilled Jockey to negate the penalties of handling, and went from there. It can take a beating (high armor and good hull), carries a sizable cargo, has decent weapons, and has plenty of bunk space. They loved a number of the other ships, but as only one player put ranks in Pilot, the group didn't like the idea of changing to two ships, especially as there were 4-7 people at the game depending on how many players were out.
For a starting ship: Space Turtle. For a ship to move up to. . .well, that's entirely based on what the group's goal is. An old campaign I ran had the group steal a Marauder Corvette with a skeleton crew that served under them, while another game had a small group of freighters based on each player's need (Firespray, YT-2400, and an HT-2200). Granted, there was only one "real" pilot, so the others were there for support (HT became a droid and explosive factory, Firespray was a fighter/bounty collector, YT was "home").
Again, it's all based on what the group needs, and needs do change.