Hello, and thanks for the question! Glad to hear you are enjoying the game so far :D
New quests come available in a variety of ways:
1) As Nolan levels up new quests will become available (try checking back in old places every 5 levels or so...)
2) As you complete the main storyline, new areas (and accompanying quests) will open up as well.
3) Once you have reached the "midway point" of the game, new quests will open up in nearly every major location.
4) As new characters join the party and move back to town, their storylines will open up as well.
Just note that while storyline and faction quests generally do not open up until you are an appropriate level to complete them, there are some requests that can be a bit difficult to complete when you first unlock them. Don't worry; there's no "timer" or anything on those quests so you can always come back later!
Based on what you've written (and avoiding any major spoilers)...There is a person inside Nyla's family store that should have a quest for you once you hit Level 10, and the Deadhander quest will NOT require you to go into any high-level areas (though the Antlion CAN be tough...). Just head in a straight line east from the outpost and you will run into the encounter you need near the edge of the large dead forest (you will find a hidden area to explore...the Antlion is lurking somewhere around that zone). Finally, when you first start out, the Trailblazers should only have one quest available (the one requiring you to gather 25 Wasp Stingers); completing it right away can be a bit grindy, but will unlock some nice low-tier gear options for your party. If you've done all that, I'd recommend starting on the main storyline (helping Gregory) and you will find by the time you return there are quite a few more things to do. If you ever are unclear about what to do next in the main storyline, Bridget will give you a reminder (she's near the kitchen table in your home in Pedregal Springs).
I hope this was helpful, and don't hesitate to ask any other questions you may have!