Yay! Another possible convert! Well if your location listed is correct (Tulsa, OK) then you're in Area V in the US. Area V has a lot of fun events, though I think most will be a drive for you (pretty much everything in Area V is a good drive away).
For the most current info on the goings-on in eventing check out USEA - United States Eventing Association (Eventing, Combined Training, Three Day Event, Horse Triathlon, Equestrian Sport)
Basically, you've got 6 standard levels in the US : Beginner Novice, Novice, Training, Preliminary, and Advanced. Some shows have baby levels that are called various things like Amoeba, Entry, Green as Grass, etc. Not all shows are 3 days. The 3 day events are either Training Level 3-Days, Preliminary Level 3-days, or are FEI reconigzed events are are "starred", for example CCI/CIC*, CCI/CIC**, CCI/CIC***, or CCI/CIC****, each of which corresponds to a US level (* - Prelim, ** - Intermediate, ***/**** - Advanced). Beginner Novice max height is 2'6" or 2'9" (I believe), Novice is 2'11", Training 3'3", Prelim 3'6", Intermediate 3'9", and Advanced 4'3", this does not include spreads, which can be quite a good deal larger than the height.
Beginner Novice and Novice are basically entry, welcoming levels. The main difference between them is height, and then you'll start to see things like 2 or 3 stride combinations at novice, maybe a jump 2 strides after going through water, etc. You'll also see some down banks as opposed to just up banks.
My suggestion to you would be to find an eventing trainer in your area that's competing higher than the level you can see yourself competing in to begin with. Now depending on your dressage and jumping experience already, that could be beginner novice or novice. I wouldn't suggest doing training to start off, regardless of your other experiences, simply because adding in cross country is a whole different ball game. What you may find beneficial (if it is in your budget) is to find a separate dressage trainer and then go to the eventing trainer for show jumping and cross country schooling.
Above all, though, is to find a good trainer that can help you safely get started so you can really enjoy yourself! After all, eventing is supposed to be fun! I've found that I prefer eventing people over those in other disciplines sometimes. Don't really know why, but I think it's because everyone goes out to a show and really wants to see everyone else do well! I've never had as many people come up and say "Good luck" or "have a good ride" as I have at an eventing show.
EDIT: Forgot to answer your other questions, lol. Most horse trials are 2-days long, not three. You'll do dressage and usually show jumping on the first day, then cross country the second day. Some places switch the jumping around so it's more traditional and you do cross country then show jump the last day. Dressage is scored the same as "regular" dressage. The score sheets even look exactly the same, the tests themselves are very similar to USDF dressage tests. Cross country is time-based. You're given an optimum time and you want to get as close to it without going over. At the lower levels there's also a speed fault time, so if you go too fast you get penalized. A refusal on course is 20 faults, a fall is elimination, and after 4 (I believe?) refusals you are eliminated. There are also, more technical faults that give you faults, but at the lower levels it's never really something you see. Stadium is judged the same as a show jumping course. You're given an optimum time, but basically you just don't want to go too slow or pull rails (or have refusals, for that matter. Rails are 4 faults, refusals are 4 faults, 3 refusals is elimination, and a fall is elimination.