there are almost no quests in this game
but you will notice when you kill ANYTHING that a message should pop up that you have killed 1 of 15 "insert mobtype here"
After you killed 15mobs of the same type it will complete, give you some xp/gold/etc... and the next step will be "kill XX of the mobtype" with X being 32 on the second step, and going up each time you complete it.
Every mobtype has it's own "quest" (called killtask here) and they reset every day.
At low levels it does not seem like much that you get, but especially in the levels from 40 to 50 they will be a huge source of XP
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Additionally you have XP-items. They drop from specific mobs and you can identify them in your inventory by looking for grey items (normally sell-loot) that have a capital letter at the front of their name. You can rightclick->shift-I them to see where you can trade them in for XP and in which levelrange you can use them.
Every item can be turned in 10times total, and each level you can only turn in 10items.
There is a list on the wiki with all (most?) availalbe XP-items and often comments about where to find them. Some locations are outdated because the frontierzones got changed now, though you might want to ask ingame for help.
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last but not least, when you solo you get Phoenix-Eggs. You can turn them in at any trainer for additional XP. Basically when you are lvl10 and kill a lvl 10 mob you will get 10phoenix eggs. At level10 you will need to turn in 10phoenix eggs and will get the XP that a level 10mob would give you (if i remember right). So don't save up too many eggs at low-lvl since they devalue a little bit. In later levels it doesn't matter that much since the difference from having to turn in 41eggs instead of 40 isn't that much of a loss. Needing 6 instead of 5 at a lowlevel is a bigger "loss". But as always, you have to weigh it against time. It doesn't make sense to make a trip to town that takes 5minutes just to turn in eggs that give you less XP than you would have gotten by simply killing stuff for 5minutes.
Feel free to ask any questions you (probably) have now
PS: almost forgot: type "/task realm" ingame and you will see a list with all mob-types and a %-number behind it. This % is the bonus XP that the respective mobtype gives. Try and hunt Mobtypes with a huge bonus, instead of ones with a low bonus