I just went through this, so here's what I learned. I'll answer your questions first.
How many of the rules of the fantasy world should be established in the first book?
Whatever rules apply to the book you're currently working on. Readers need to understand how your world works pretty quickly, though fantasy readers are more patient about this since they know it takes time to build a good world. But any critical rules need to be there early on.
Should the reader be given most of the rules so they will know how the world works before they move on?
If they're important to understanding the events driving the plot, yes. If the rules don't affect the plot, no. You don't have to explain the entire world right away, since most likely there's stuff that isn't relevant to the first book.
Each book needs to be it's own story, especially book one. Establish the world and the rules that are critical to understanding your protag and their current problem. Make that current problem clear and have something the protag is trying to resolve in book one. There can be a larger story arc that spans all three, but the first needs to be able to be read on it's own and still be a satisfying read. They shouldn't be working all book to solve a problem and not solve it. Let them at least solve the first step of the bigger arc, and let book one be all about that.
The next two books should also be able to be read on their own, even though there will be some things that new readers won't fully get. Like you need three major steps in the story arc. One per book. One story broken into three parts is an almost impossible sell for a debut author.
The middle book of a trilogy is one of the hardest things to write. It's a lot like the middle of a novel, where things tend to bog down. It's often ends up a set up for book three, so it doesn't feel like its own book. A solid "stand alone" plot here will help combat this a lot. (I rewrote my book two five times before I got it right). The plot can still be part of a larger story, but there's something specific that needs to be done that's driving your story, and that's resolved by the end of book two.
Trying to figure out what back story and info from the previous books to put in will likely drive you nuts. I found using only what was relevant to the current plot helpful in knowing what to put in. You'll also need to remind readers about any world building and world mechanics and how that works. (and do it without re-doing what you did before. This can be a challenge finding new ways to show this). I also found that thinking about book one as "backstory" and looking at the goal your protag has for book two helped a lot in keeping me focused.
Character growth is important as well, not only for your protag, but for the secondary characters. Second and third book are great places to develop the best friends and sidekicks. Be wary of adding too many new characters. Only add ones that are critical to the next step of the story. I found recycling old characters very helpful here. Throwaway ones or very minor ones could fill bigger roles in later books.
Stakes need to continue to escalate, but be wary of starting book two off with too high of stakes. This is easy to fall into if you treat it like the next chapter of book one, since book one ending with super high stakes (or it should have). New information about your characters, the world and the problems also need to be revealed over the course of the trilogy. There has to always been something for the reader to discover.
You don't need to have the full trilogy written before you submit to agents. A one or two-page synopsis for the next two books is enough for your agent to sell the full trilogy. (that's what my agent did). However, if you're in no hurry to submit, having all three done so you can edit book one if you think of something really cool while writing book three has its advantages. There's stuff I wish I could go back and add to one that I can't since it's already on the shelves.