All right. I'll start with ships.
There's two new post-50 classes, T5-U and T6. T5-U's are upgraded versions of existing T5 ships, or a number of new ships available from lockboxes released since DR, which start as T5-Us from the word go. You upgrade your existing T5 ships with Upgrade tokens from the C-Store. They're not account bound, so you can often find them on the Exchange as well. Depending on price, that might be the best option. Once you have the token, you can upgrade them at the ship status screen. Now, I'm not sure, but you might have to talk to the ship selectors and do it there. I will note that all event ships (The Breen ships from the winter event, the Risa ships, anniversary ships and such) can all be upgraded for free without a token, and the mirror universe ships can't be upgraded at all.
Now, there's three differences between T5 and T5-U. There's a boost in hit points, that improves the closer you are to level 60. There's an extra console slot, the type of which (Tactical, Science, Engineering) varies from ship to ship. And there's starship mastery. Basically, in space combat, the ship earns XP for itself. T5-Us ships have four levels of mastery. Each level grants a different buff to the ship, that varies by ship. More HP, turn rate buff, accuracy bonus, carriers often get buffs to their fighters, and so on. The levelling for those is fairly easy, you'll manage that pretty quickly without any trouble.
Frankly, if you want to keep flying existing ships, the T5-U upgrade's your best bet, although I've heard complaints that not all the old guard really work out that well at level 60.
T6 ships are every new C-store ship released from Delta Rising onwards, along with the new Event ships, which you missed out on, so just C-store ships I suppose (Speaking of Events, there's some stuff there I'll get to later. It's kinda nifty, bur also kinda annoying). T6 ships have the same hp boost that T5-Us have, maybe slightly more, although they don't have an extra console slot. They have a console, almost always usable on only those ships, which are pretty powerful. There's two and a half big differences that separate them from the 5-Us.
First, their Starship Mastery tree has five levels, not four. The fifth level is actually a starship trait. Once you've unlocked it, you can activate it in the traits window under ship traits. Like the space and ground reputation traits, you can have four active at any time, and once you've unlocked them, they apply no matter what ship you're flying, be it a 6, a 5, a 5-U, or even the NX replica class level 1-10 captains can fly
The trait varies by ship, and I'd recommend looking up various ships on the wiki to see if there's any you really need.
The other difference is that all T6 ships have access to one of the Specialisation trees in some way. You can now give any BOFF a secondary class in addition to their base one, currently Intelligence, Command and Piloting. Ground abilities in those trees can be used under any circumstances. Space abilities are limited to certain ships. For instance, all the ships that came out with Delta Rising have an Intelligence slot. The recent Command and Pilot ships have similar stations. Those ship slots are dual-class, and vary depending on the type of ship. For instance, one of my captains has a ship with an Engineering/Intelligence slot. You can slot any Engineering officer, and if they have Intelligence skills, you can use those as well. The new powers vary, some are impressive, some are situational. I don't think they're gamebreakers, but opinions might vary.
Lastly, there's the 'half' difference I mentioned. A number of T6 ships, namely the Intel, Command and Piloting trios, have special abilities locked to that sort of ship. They're a lot like Sensor analysis, Cruiser Commands, and Romulan Singularities. Intel ships can send out drones to apply debuffs to enemy ships, Command battlecruisers can buff their teammates, and Piloting ships get afterburners, barrel rolls, and other fancy moves. Again, I don't think they're gamebreakers, but they're fun.
(Actually, I lie. Command battlecruisers ARE gamebreakers, in that their powers have been doing something WEIRD to the servers. It's to blame for a lot of the lag problems, and I know the Devs are reworking the background stuff to put a stop to it. Oddly, the other big offender is Attack Pattern Beta...)
So, yeah. T5-U for the classic ships with some minor bonuses, T6 ships for the new hotness. Although they've started making T6 Iconic ships that can use some of the older costume parts. There's a T6 Galaxy/Venture class, for example.
There's two new post-50 classes, T5-U and T6. T5-U's are upgraded versions of existing T5 ships, or a number of new ships available from lockboxes released since DR, which start as T5-Us from the word go. You upgrade your existing T5 ships with Upgrade tokens from the C-Store. They're not account bound, so you can often find them on the Exchange as well. Depending on price, that might be the best option. Once you have the token, you can upgrade them at the ship status screen. Now, I'm not sure, but you might have to talk to the ship selectors and do it there. I will note that all event ships (The Breen ships from the winter event, the Risa ships, anniversary ships and such) can all be upgraded for free without a token, and the mirror universe ships can't be upgraded at all.
Now, there's three differences between T5 and T5-U. There's a boost in hit points, that improves the closer you are to level 60. There's an extra console slot, the type of which (Tactical, Science, Engineering) varies from ship to ship. And there's starship mastery. Basically, in space combat, the ship earns XP for itself. T5-Us ships have four levels of mastery. Each level grants a different buff to the ship, that varies by ship. More HP, turn rate buff, accuracy bonus, carriers often get buffs to their fighters, and so on. The levelling for those is fairly easy, you'll manage that pretty quickly without any trouble.
Frankly, if you want to keep flying existing ships, the T5-U upgrade's your best bet, although I've heard complaints that not all the old guard really work out that well at level 60.
T6 ships are every new C-store ship released from Delta Rising onwards, along with the new Event ships, which you missed out on, so just C-store ships I suppose (Speaking of Events, there's some stuff there I'll get to later. It's kinda nifty, bur also kinda annoying). T6 ships have the same hp boost that T5-Us have, maybe slightly more, although they don't have an extra console slot. They have a console, almost always usable on only those ships, which are pretty powerful. There's two and a half big differences that separate them from the 5-Us.
First, their Starship Mastery tree has five levels, not four. The fifth level is actually a starship trait. Once you've unlocked it, you can activate it in the traits window under ship traits. Like the space and ground reputation traits, you can have four active at any time, and once you've unlocked them, they apply no matter what ship you're flying, be it a 6, a 5, a 5-U, or even the NX replica class level 1-10 captains can fly
The trait varies by ship, and I'd recommend looking up various ships on the wiki to see if there's any you really need.The other difference is that all T6 ships have access to one of the Specialisation trees in some way. You can now give any BOFF a secondary class in addition to their base one, currently Intelligence, Command and Piloting. Ground abilities in those trees can be used under any circumstances. Space abilities are limited to certain ships. For instance, all the ships that came out with Delta Rising have an Intelligence slot. The recent Command and Pilot ships have similar stations. Those ship slots are dual-class, and vary depending on the type of ship. For instance, one of my captains has a ship with an Engineering/Intelligence slot. You can slot any Engineering officer, and if they have Intelligence skills, you can use those as well. The new powers vary, some are impressive, some are situational. I don't think they're gamebreakers, but opinions might vary.
Lastly, there's the 'half' difference I mentioned. A number of T6 ships, namely the Intel, Command and Piloting trios, have special abilities locked to that sort of ship. They're a lot like Sensor analysis, Cruiser Commands, and Romulan Singularities. Intel ships can send out drones to apply debuffs to enemy ships, Command battlecruisers can buff their teammates, and Piloting ships get afterburners, barrel rolls, and other fancy moves. Again, I don't think they're gamebreakers, but they're fun.
(Actually, I lie. Command battlecruisers ARE gamebreakers, in that their powers have been doing something WEIRD to the servers. It's to blame for a lot of the lag problems, and I know the Devs are reworking the background stuff to put a stop to it. Oddly, the other big offender is Attack Pattern Beta...)
So, yeah. T5-U for the classic ships with some minor bonuses, T6 ships for the new hotness. Although they've started making T6 Iconic ships that can use some of the older costume parts. There's a T6 Galaxy/Venture class, for example.