GW2 is, at its core, a combination of what many consider the best ideas in MMO gaming across the last decade.
There are no set quests that you pick up and then go do like a traditional MMO (the exception being the personal story quests for your character). Instead, PvE areas have places indicated on the map by Hearts, which will provide you with a goal once you get close enough to the spot. The goal is a bar that fills up the more you help out whatever NPC(s) require aid. Usually there are a couple ways to help, which might include driving off pests, harvesting meat from something, rounding up cows, etc. Similar to the range of things you might do on a standard quest in another MMO.
In addition, all around the map, dynamic events spawn that you can help complete if you want. These are often combat oriented, but not always, and the events scale depending on how many people are there participating. It's kind of like the rifts in Rift, except you'll usually find yourself defending an outpost, raiding a bandit lair, or killing a boss, etc. Some of the dynamic events are chains and completing one will spawn another. Every area has an overall boss that, when the chain is completed to get to it, spawns and offers a large fight for those interested. You can ignore any of these things, including the Hearts, by just not doing them - but it's how you gain exp and level.
PvE also includes exploration, and you get rewards for finding all the points of interest, waypoints, etc on a map (and doing all the Hearts). Most areas have at least one jump puzzle, and some of them are seriously long and have some thought put into them. Some of them are easier to find than others.
Each class can use certain types of weapons, and what weapon you're using determines your attack skills (number keys 1-4). You have to acquire each type of weapon and use them to unlock those skills, but it doesn't take long to do it. This allows you to switch up your role pretty easily (not once in combat, though). So one might allow you to be more a ranged support, while another a melee combat, and another a ranged dps, etc.
There is no dedicated trinity (tank, dps, healer) in GW2. Every class can do a bit of everything, and every class has some healing they can do to support themselves. Certain classes can switch into more of a healing/support mode, though. You're able to unlock and slot other abilities based on your class as you level up, and these help customize your current roles. They have a wide variety of uses, and can include summon pets to help you fight, turrets to kill stuff, healing circles, damaging skills, debuffing things, buffs, traps, snares, etc. Through your weapon and what types of skills you have slotted, you can setup your character to do what you want.
Combat tends to be more fluid than something like WoW. Dodging plays an important role. I've seen people say dodging is useless except for avoiding major, telegraphed attacks. To some extent that's true, as you do absolutely need to avoid the huge attacks, but dodging other smaller attacks when you're able will still make you more effective in combat. I approach PvE combat in GW2 more like PvP combat...always moving, always dodging.
PvP auto scales you UP to level cap, and takes place in large areas with a lot going on. There's a lot to do if you want, other than just standing around trying to kill other people. It's a WvWvW setup, so it's like a competition between servers. And then of course there are dungeons starting at lvl 30. Without the trinity, they work a little differently than normal dungeons are often a bit more chaotic rather than controlled. To do some of them on the harder difficulty, though, you need to sort through some of the chaos and manage the fights, just like any other MMO.
It's one of the better MMO's out there, at least to me. But even though it's not everyone's cup of tea, it's one of the few MMO's I'd recommend everyone at least try.