So pretty much everything will give you XP. Crucible matches. Completing Story Missions (only the first time you do them via the map, then again for the "Daily Heroic"). Strikes (only the first time via the map, then again each time in the "Vanguard" Strike playlists that unlock for you at higher levels). Killing enemies. Bounties.
XP will level you up all the way to level 20, then the system changes and XP won't directly effect your level.
It works kind of weird though, because if you complete a bounty for say 5000 XP, you'll earn 5000 XP toward your level, and then 5000 XP for each item you have equipped, rather than it splitting the 5000 between everything. That is, you'll earn 5000 XP for each of your equipped weapons and armours AND subclass, so make sure you've equipped whatever gear you are interested in leveling when earning XP or turning in bounties.
Not everything is available at the beginning of the game. Certain things you won't even see until you've reached a certain level and then the game will give you a sort of notification letting you know that you can now access it.
In Destiny, there are Factions. Each Faction has at least one NPC in the tower where you can purchase their corresponding sets of weapons and armours and there are different criteria for leveling them and currencies for each.
The Factions (so far) are:
Vanguard
Crucible
Cryptarch
New Monarchy
Dead Orbit
Future War Cult
Iron Banner (only open for one week each month)
Eris Morn (The Dark Below NPC)
Vanguard is in charge of all PvE content. Crucible, PvP. So if you complete a Vanguard bounty, you earn "Reputation" (different from XP) toward leveling up Vanguard faction. Same with doing Crucible bounties toward the Crucible faction. There are a couple other ways to level up factions at higher levels as well once things start to unlock for you.
Each faction (Vanguard, Crucible, New Monarchy, Dead Orbit, Future War Cult, Iron Banner, and Eris Morn) requires that you be a certain "Rank" in that faction to purchase their corresponding gear. For most, it is Rank 3 for weapons, and Rank 2 for armours.
In addition to Reputation, Vanguard and Crucible both have "Marks", which is a type of currency you can earn by successfully completing certain activities. You earn around 5 Crucible Marks for winning a Crucible match, for example. And you earn some Marks for Vanguard by completing a Strike in the Vanguard Strike playlists (which unlock at higher levels). You can earn a maximum of 100 Marks per week, and you can hold a maximum of 200 Marks total (of the two types).
edit: Also, in regular Crucible (that is, not Iron Banner) your weapons' Attack values and your armours' Defense stats are all flattened and are not calculated, so you aren't at a disadvantage for having a low-level gun or what have you (aside from the perks on some of the higher level weapons or gear)
Once you hit level 20, your level will be tied to the Light stat of whatever armour you're wearing, which is why people tell you to rank up your Vanguard, Crucible, or whatever factions early. You won't be able to rank up New Monarchy, Future War Cult, or Dead Orbit until level 20, though.
The above is only the beginning. Experiment and play around with how you like to play your character, what class of weapons you prefer, etc. You can't really do anything wrong as most of what you do will progress you in one way or another. You will hit Level 20 before you earn enough XP to get all the abilities in your subclass, too, so there'll be a lot of variety in how your particular build plays. You'll feel more powerful too as you unlock new "perks" in your subclass that enable you to do increasingly powerful things. Destiny sort of starts you off with the basics and you feel kind of like a Spartan from Halo at first, but as you level, you'll get more and more access to more and more abilities. That's the whole "Become Legend" thing
I didn't cover everything, but that's the basic stuff for now. Hitting Level 20 and ranking up either your Vanguard or Crucible to level 2 early will make your life easier later though.
Oh ONE more thing, you'll notice some enemies have different coloured shields. There are 3, Blue (Arc), Purple (Void), and Orange (Solar). You can pop those shields with any type of damage, but they are particularly weak to weapons that do the same colour of damage as their shields. You'll start finding weapons that do one of the 3 types of damage pretty soon (Special and Heavy mainly). Certain enemies also do certain types of damage if you look at the colour of their bullets. Same goes for your special abilities like grenades and supers, too.
I hope that's not TOO confusing (or obvious!), the game does sort of overwhelm at first but it's worth it, in my opinion
