What transfers with your character in New Game+ mode?

If I decide to use New Game+ after completing the game, what transfers to the new game?

Will I have an opportunity to try a new character class or gender?
Will I be locked into the same Shepard that I chose originally?

Edit If I originally started with a save game from ME2, will the choices I made in ME2 still be in effect for the New Game+ play through or will it use the ME3 standard set of choices for a new game?

No, you cannot change your gender or class on a NG+. The only customization options are the face and the bonus power (Any 1 power you've unlocked). Your credits, reputation, unlocked weapons, armor, weapon upgrades and weapon mods transfer over, as well as collectibles like fish and the hamster, through some weapon mods may be downgraded 1 level. Your choices from ME1 and 2 will also remain.

Picking up weapons you already have will upgrade them 3 levels.

If you wish to change class or gender, you can use a save editor.

Open the NewGamePlusSave.pcsav from your completed playthrough, make your edits, and save as a new career to import for your NG+.