Early: Apogee/Heron/Aurora
You and I must have vastly different opinions on what constitutes "early game" if you're throwing the Aurora in there. You can start with an Apogee and can find a Heron relatively quickly but I've never found an Aurora (bought, salvaged, or otherwise) until late-game, if ever.
As far as vanilla goes:
Early:
Hammerhead is my go-to, usually at medium or short ranges. Heavy Mortar + Heavy AC is a pretty good combo that isn't too hard to manage but I prefer the Assault Chaingun with Heavy Machine Gun if I can manage it. If I can find a Tempest (which isn't terribly hard), a Phase Lance and Graviton Beam is a great early combo to wipe out Frigates and really pressure Destroyers.
Falcon (P) is also a fantastic early game ship, if you can outfit it properly. It really punches above its weight in small-to-medium fights and it doesn't slow down your fleet of Destroyers.
Mid-Game:
I tend to eschew capitals until I can support them so a solid Cruiser is where I gravitate in the "I have a colony but still need to fight bounties for cash" phase. If I can get an Aurora, that's my unicorn but those are really rare. I really like SO'd Eagles with Assault Chainguns and Heavy Machine Guns, sort of like a larger, more aggressive Hammerhead. It lacks the pure offense that Accelerated Ammo Feeders has but the Eagle is a much sturdier platform and has much more flux to spare so the Energy weapons can be AM blasters, Phase Lances, Ion Cannons, etc.
Late-Game:
If I have well-oiled colony machine and am making money hand-over-fist, a fleet with 3-4 Capitals isn't out of the question for me. The Odyssey is my go-to because I prefer the mobility and because it can hit hard for a very long time if built right. Legions are also about in the same boat but their hard to justify from a logistics standpoint.
If I can find an Aurora, that still tends to be my favorite because I can make it flux neutral and it
flies all over the battle space. It's faster than most frigates. It does lack in the heavy-hitting department but its so slippery that against everything but a Star Fortress, it can get in and out without much trouble. The range of stations make it less than ideal.