Lots of good advice in this thread already, here are mine, from a pve-only perspective.
Leveling 1-20:
-All classes are fun, pick whichever one you seem to like best. A Gentleman plays a Warlock.
-don't do missions over your level unless you have to. The damage penalty will slow down your progress.
-do bounties, good xp and they'll keep you over the level of your story missions.
Post 20:
-specific activities will give you currency, called marks (vanguard marks for pve and crucible marks for PvP). Daily missions, weekly missions, strike playlists, random events during patrols, and PvP.
They can be used to buy legendary gear, which is the most reliable way to level up.
-These activities, as well as the bounties, will give you reputation with either the vanguard (pve) faction and the crucible faction. You need a certain level of reputation to access to the store items.
-if you like the game, keep whatever legendary armor you get that isn't for your class, and make alts! They become a lot faster to level, and let you do weekly/raids multiple times per week.
-don't hesitate to ask for help on the OT.