There are two core things you can do to gain experience points (XP). The first is finding things to trigger lore events, and this is generally the easiest. This is a long list, and on the RoB wiki:
http://ringofbrodgar.com/wiki/Experience_Points Characters can get stuck on some of the odder events, though, and sometimes you just need time or persistence to get past this barrier.
The second is doing quests. The rewards are frequently XP or learning points, though other bonuses from completing them exist and often get in the way when you're after the XP.
If you're a part of a realm (as a member, not just living there*), you can ask the leadership to give you some XP out of the realm's authority pool. (Realm authority is generated when players earn XP.) There may be costs involved, it might be given freely.
You can also set up an XP farm of sorts, and most of the established villages or realms have one. This is an area where you can test out as many lore events as possible without a lot of travel, so if you're stuck, maybe you can get unstuck. Larger realms or villages with public areas might have one that any player can use, though do to the permissions system, it wouldn't be as complete as you could could potentially build on your own.
As a way to conserve, avoid using magic spells if you can unless you've built up an XP buffer for 5-10k or more. Avoid using high cost (XP/hour of study time) curios unless you have the same buffer.
Droughts of XP are common for everyone. It's learning how to manage it properly that is the trick.
*It's been a while since I've looked this mechanic up, and can't seem to find anything other than this. Please correct me if I'm wrong. Thanks in advance.
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