A lot of new players around so: The basics of GW2 and how it's different from other MMORPG
Not quests but Dynamic Events in PVE
In GW2 the core of PvE gameplay aren't quests, but Dynamic Events. I'll try and explain a bit just what this means. In most MMO's you get quests (usually in a hub) for your level, then grind some mobs and after that you do the next batch of quests. In GW2 things happen all around the the world and you have to be there to participate. If you don't complete an event, somebody else probably will. The gameworld doesn't wait for you to push a button, it moves on its own. You need to keep an eye out and jump into the action whenever you can. You can also ask in map chat if any events are up and usually somebody will link a waypoint (fast travel system, very low fee) so you know where to go. Due to the downlevel mechanism, all the maps and their events stay relevant to the max level players. Even if you do events in the low level starting zones, you'll get the rewards of your actual level.
There are over 2000 dynamic events in the game. I played since release and I still haven't seen them all! You can repeat an event as many times you want. There are some events that I have done several times an evening, just because I like them. The rewards you get for completing an event are always the same, just scaled to your level and contribution. The events are also designed that anybody who sees them can join in on the fun. You can do a lot of events with just two players, or call in reinforcements through map chat. Whatever you want. Events also happen in a chain, meaning that usually 5-10 events follow up on each other. When an event is finished, stick around. Clear out your inventory for example, talk to the NPC, gather that tree just a bit away ...
Largely there are three tiers of events. The normal ones happen all the time and you can sometimes even do them solo (if you are trained :D ). The group events are usually the last event of a chain and are designed for five people and more. Then we also have the World events that are designed for dozens of players. These events happen on a fixed timer that you can find on internet (search: that_shaman dulfy.net timer). The game gives you several hints to get to events. The biggest are orange circles on the mini map and popping up in your UI. You have to be quite close to see these indications. There are also NPCs that run towards you and ask for help. If you talk to them, they can point towards an event. Some events also have an audio message that signals their start. It also happens that an NPC has an orange circle just above their head. That means they are an event starter: talk to them to get the show on the road.
Even apart from dynamic events, almost everything you do gives you xp. The max level is 80. In Casual play that will take about 100 hours to reach. If you power-level (not at all recommended on a first playthrough) you can get 3 levels an hour.
Combat without the holy trinity as you know it
The holy trinity doesn't really exist in GW2. So there are no tanks or healers, period. No profession or build can just stand in front of mobs and soak up the damage without being a dead weight. Taunt exists, but is very limited to a few skills. The same counts for outhealing damage since it's more important to avoid damage. Everybody has to block, dodge, kite, blind, interrupt, line of sight, reflect, ... A build could maximize it's ability to kill mobs, controlling the battle field or supporting the other players. However, most of the time a build is a mix of all three. The aggro system doesn't follow clear lines. So nobody can just take are lose aggro. Some mobs will focus the player who is closest, or the furthest, or has the highest toughness, or is downed... There are mobs that lock on to one player and keep attacking him until he is dead. Most will regularly switch targets. And so on ... a lot of mobs use an infrequently big attack that you are just meant to avoid. The have to watch their animations so you know when to dodge. This is especially true for bosses. Almost every boss has an attack that can one-shot kill almost every player.
Everybody is primarily responsible for his own survival. You ALWAYS have your own healing skill on skill slot 6. People can help each other out with area heals and boons, but when you die you usually could have done something to prevent it. Every profession can fight from ranged and in melee. Melee is almost always advised, because it has higher damage across the board. It's also a lot easier for people to support and buff each other.
In PvP we do have bunkers. These are builds that focus on staying alive under pressure but not just by maxing certain stats. Position, dodging, blocking, healing, interrupting are still essential for a bunker. The same goes for WvW.
In PvE, people usually focus on berserker gear, that power/precision/ferocity (damage+crits). That's because this gear does the most damage in a short amount of time. It basically means kill fast, or be killed yourself. If you can't survive in full Berserker gear, mix a couple of Soldier (power, vitality, toughness) pieces in. The most common thing that kills new players is that the level of their gear is too low compared to their actual level. This is especially true when you get downleveled, since the system increases the disadvantage of underleveled gear. Ideally you should upgrade your gear every five levels. Usually you will get loot drops that keep you going. You can also buy it on the trading post for very little cost and when you outlevel that gear, you can salvage it to recover most of their worth. While leveling, you should only look for fine (color blue) or masterwork (color green) gear. Rare (yellow) and Exotic (orange) are too costly for the amount of time you'll use them.
In WvW (World vs World, a massive PvP game mode between hundreds of players over 4 maps) the standard setup is soldiers gear (that power/vitality/toughness). If you're going into WvW with a lower level character, you're going to have a hard time. In WvW you can really die before you know what hit you, especially if you aren't max level with end game gear (meaning exotic or ascended).
Structured Player vs Player
You can start PvP as soon as you exit the tutorial and you will immediately get access to all the skikls, traits, weapons and have the same possible stats as everybody else. A max level character doesn't have any advantage over a level 1 in PvP. SPVP has it's own 'build' menu where you can decide how you want to play. All the stats from gear are given by an Amulet you pick for your build. The only overlap between SPVP and WvW/PvE are the type of weapon you have in your hands (again, stats for SPVP are on the Amulet). It's completely possible to level up a character solely by playing SPVP because of the rewards you get. Those rewards are grouped in the reward tracks. If you want to get high end gear that you can use in PvE/WvW, it's best to do the Dungeon reward tracks. However, you will have to do the story mode of the dungeons in PvE to unlock these tracks, or you have to wait for the weekly rotation of dungeon tracks. Most SPVP is three capture points combat in teams of five. You can join a game on your own and will be matched with other players, who hopefully are in your level. The three capture points are designated as 'close', 'mid' and 'far' respective to your base. Your team gets 1 point each second it controls a capture point and 5 points per enemy kill. First team to 500 points wins. Other game modes are in development and will probably ship with the expansion.
World vs World, the large scale pvp
This is the large scale open world PvP mode of the game where people play for their server against two other servers. It's considered end game, because lower level players are actually a disadvantage for the server. Lower level players get upscaled, but not enough to be evenly matched. WvW is also capture point based. Every 15 minutes the timer ticks and each sides points are added to the overall score. Matches last for 7 days. Each server has its own home borderland and then there is also the Eternal Battlegrounds. A big part of WvW is 'capture the tower' with siege. Killing enemies gives the most loot, but not the most points. Defending is crucially important, but very unpopular.
What to do, what is the end game?
Actually the game is what you want to make of it. You don't really have to do anything. There are over 30 explorable dungeon paths, personal story, living story, world bosses, crafting, six mini games, pvp, wvw, 12 fractals, over 30 jumping puzzles, a lot of hidden areas, collectible achievements, guild missions, guild events, community events, festivals, hidden treasure chests ... and those thousands of dynamic events. GW2 is a living world, things happen all the time and you can miss it if you're not there. There are lots and lots of options to have fun, but you have to go look for it, you have to find out for yourself what you want to do. The game doesn't hold your hand and point to where the action/fun is. It's your game, it's your time, it's your move. Just take your time, relax, try everything a bit and have fun. Maybe you'll even find a goal you never though of at this moment. I know of people whose goal it is to solo all the champions in the game. People who want to have 40 level 80 characters, one for each combination of race and profession. Others compete to have the fastest dungeon run and upload their playthrough videos with timers on special sites. People that want to have every legendary. Some even go for world completion with every profession. Others solo the party dungeons.
For many people their look, minipets, dyes, title ... are also extremely important which gives this game the nickname Fashion Wars 2!