I tend to pilot the biggest ship in my fleet, unless it's a carrier or an "anchor" (i.e. a Dominator in an otherwise Destroyer-heavy fleet).
Early game, I really like the Hammerhead as a general go-to but if I can find a Tempest, it's about the only frigate I'll consistently pilot over a Destroyer. It's extremely fast and can punch above its weight using HEF and a Heavy Blaster/Phase Lance+Graviton.
Mid-Game (pre-colonies), an Eagle is a really solid ship and with SO, it can push out some ruinous firepower while remaining very nimble. The Falcon (P) is also extremely powerful in the player's hands. I'll probably have a few Cruisers at this point but I generally can't find the top-tier ones like the Aurora or Doom, otherwise those would be my #1 choices.
For the endgame, I prefer the Odyssey most. 2x Plasma Cannon is absurd, it's as fast as a Cruiser, isn't a logistical nightmare and doesn't slow my fleet down. It gets into a little trouble against stations (where mobility doesn't really save it) but can otherwise go toe-to-toe with about anything. The Conquest is also good but I find it to be less versatile than the Odyssey, overall. Certain Onslaught builds are fun if you just want to facetank everything but I find that to be a little less rewarding than the positioning battles that the Odyssey can get into. Finally, there's the Paragon. I have a love/hate relationship with it: I love it because I can not only outgun anything fielded against me but I can outlast it, too. It's this behemoth of a ship that can annihilate whole fleets but unless I want a power trip, it's not super-engaging.
So, to summarize, it depends.
I think my favorite "class" is the battlecruisers followed closely by the top-tier Cruisers (Aurora/Doom). I think I will really like the new Champion and am optimistic about the Fury. I don't particularly like the Shrike but the Fury has a bit more firepower and will probably play similarly.