"Class" is a slightly loose term
In character creation, you'll have to make a choice. Do you want to be really good at ground combat or space combat? Maybe you want to be a little good at both, and that's an option. However, many of the traits you can select from are geared towards emphasizing one way or another. Warp core theorist, for instance, will help with your ship's energy levels -- a useful thing to have when you're trying to reroute all power to shields or weapons.
Another thing you'll need to carefully consider is your captain's chosen profession. While you can pilot any ship regardless of your captain's specialty, your chosen profession will decide where you're at your best.
For space, Engineers specialize in Cruisers, which are meaty mammoths equipped with large crews and sizable weaponry. Escorts are smaller, faster and much more heavily armed ships that dish out far more raw firepower than they can take. Lastly -- but of course nearly as important -- are the Science ships, which support allies and weaken enemies. Science vessels also come equipped with cutting edge sensors, making them extra useful when dealing with, say, sneaky enemies like the Klingons and their cloak tech.
As for ground combat, your role is more set in stone. The big thing here is your "kit" which comes with a special ability you can use during combat. Engineers can equip a kit that lets them teleport small phaser turrets or shield regeneration stations. Tacical kits allow you to toss photon grenades or temporarily boost your ranged capability.
There's no XP!
Technically, there aren't any "levels" either. Although, there are ranks beginning from Ensign (a rank you only possess during the turtorial) and moving through Lieutenant, Lieutenant Commander, Commander, Captain and finally Admiral. Each official rank has ten "sub-ranks" a player needs to level through before they attain the next rank -- that makes for 50 levels.
Getting back to the experience matter. Well, there really isn't any, as the requirement for leveling up is attaining a set amount of skill points instead. You earn skill points primarily via PvE and PvP missions and kills. There's a randomly generated exploration mission system as well, known as the Genesis system. Oh, and there's diplomacy missions sprinkled throughout PvE as well. Remembering to spend these skill points is key, as your skills improve both your ground and space combat performance for you and your bridge officers. Just keep an eye out for the "Skill up!" button that appears on your progress bar when enough skill points are available for you to spend on new skills or skill ranks.
Dynamic duo plus
Yes, this is an MMO, which is exactly why this is a tip. You don't have to play with a large group of people, just one other person -- trust me, it makes things easier. Not only will things move along faster, but you'll have a whole lot more fun shooting lasers in space with a good friend.