Daily Quests can be found be clicking on the profile pic on the bottom right of the main menu.
These will be things like 'win five games with free masters', 'play 20 spell cards' and so on. These don't need to be done in one sitting and can be completed over several days. You can't have more than three active at any time and they won't renew until a day has gone by after completed.
Anything involving ranked play will work (so team battles, conquests, expeditions and drafts all count)
There were achievement tracking issues with this around patch 1.3/early 1.4 so if you had issues previously, it's worth checking you are fully up to date with patches. Remember as well that achievements need to be claimed (separate tab on that same profile page)