Combat is mix of WoW and Diablo. Regular attacks work very similar to Diablo, you click on the monster and your player attacks it. One click per attack. There are cooldowns after each attack, but for most weapons they are very short, half a second or so. Only two-handed, big weapons and ranged weapons such as bows and arrows have noticeable cooldowns. Cooldowns for these weapons are introduced to compensate their increased damage and/or range. Cooldowns are also affected by player's skills, active spells, other effects such as monster auras etc.
Spells work more like WoW spells. Each time you click on the monster, that monster is selected. Click on the spell icon casts that spell toward the monster. This works for missile spells. There are other types of spells, for examples protection spells which work on player and do not require target, or ward spells which create magical wards on the ground beneath the caster.
Skills also play big role in the combat. There are separate skills for all major weapon groups which increase your chance to hit monster. Also, there are “Lore” skills which show how experienced you are to fight creature of specific type (undead, animals, spiders, reptiles etc.). You can use Bestiary panel to search database of creatures you have encountered. As you fight more creatures, you will get more information about them. For example, when you kill single spider, bestiary will show it's level, health and faith (equivalent to mana). However if you kill 50 spiders, bestiary will display spider's ability scores, resistance, attack/defense scores, damage and more.
Actually there are many non-combat quests in the game. There are many NPCs that will ask you to complete various quests such as those where you have to find items or carry items to another NPC, those that require you to find or explore specific locations, those where you need to talk with another NPCs etc. Some of these quests are in towns so there will be no combat in those quests, however there are some quests that require you to travel through wilderness or to go to locations that are filled with monsters. Even though those quests are non-combat in nature, it may still be required to kill few critters just to get where you have to.
Some of these quests can be finished in only one way (for example quest that requires you to find specific location). Other quests can be solved in multiple ways. For example, if quest requires you to bring herbs to NPC, you can solve that quest in more than one way. That includes finding the herbs in various chests (for example in dungeons), or you can use herbalism skill to collect plants in wilderness, or search warehouses in the city, you can buy herbs from NPC merchants or you can even try to solve some other quests that give you what you need as reward.
Each solved or failed quest modifies your reputation with factions. Reputation affects way NPCs act toward you. Reputation also modifies prices with merchants of that faction. It's worth mentioning that some quests require you to choose way to solve quest that will positively affect your reputation with one faction and negatively with another faction. Factions in the game are not always in peace with each other, so solving quest for one faction may reduce your reputation with other one. Generally speaking it is best to try to keep every faction as happy with you as possible.
Another type of quest (which is actually single game-wide long quest) is quest for artefact pieces. Dawn artefact is broken in many peaces and in order to finish the game you will have to find all of them. Pieces are scattered throughout the game and usually well guarded - both with monsters and with puzzles. Finding these pieces is also one of the 27 achievements player can solve. Achievements are divided in three categories - exploration, combat and crafting. Each of those three categories has nine different achievements. Solving specific achievement brings you additional experience.