Your squad in ME3 consists of:
The Virmire Survivor, which means one of Ashley or Kaidan, as well as the return of Garrus, Tali, and Liara. In addition, two new members join the squad, James, and a newly corporeal EDI. If you own the From Ashes DLC, there is also Javik, the Prothean.
As for 'classes', the squad members don't have classes in the sense that Shepard does. Instead they each have their own unique set of 4 powers. Two are shared with one of the available Shepard classes, and two are unique to them, but can be unlocked as Bonus Powers through conversation.
Specifically, the lineup of powers for each looks something like this:
Disruptor Ammo Concussive Shot Inferno Grenade Marksman
Reave Barrier Overload Cryo Blast
Incendiary Ammo Carnage Fortification Frag Grenade
Concussive Shot Armor-Piercing Ammo Overload Proximity Mine
Singularity Warp Stasis Warp Ammo
Tali'Zorah vas Normandy
Defense Drone Energy Drain Combat Drone Sabotage
Incinerate Overload Defense Matrix Decoy
Dark Channel Lift Grenade Pull Slam