Welcome to the hobby!
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~What sizes are there, and the terms for them?
Every company has different names for their sizes, but some people use Volks' names, which are...
1/6 scale - YoSD / Yo Super Dollfie
1/4 scale - MSD / Mini Super Dollfie
1/3 scale - SD / Super Dollfie
1/3+ scale - SD16, SD17
Other companies have smaller dolls--like Fairyland's Puki lines--and I think they're mostly just called "tinies". Sometimes, you'll get people who get upset if you use Volks terms for non-Volks dolls, but they're arguably the most recognizable names.
~Does each doll have a certain name, or is it sort of a line of dolls?
It depends on the company--each company usually names their line based on body type, and then each doll is named for the head sculpt.
~What Is a "face-up?"
A "face-up" is what you call the makeup painted on the dolls.
~I was thinking of looking at Obitsu & Luts, good idea or no (anything cheaper of the same quality)?
Obitsu aren't resin, but can be good for starter dolls--they'd just be off-topic in some BJD groups. I believe they also have a different joint system. Luts is an awesome company, but they're pretty expensive. I'd argue they're worth it, though. The Luts Delf line is gorgeous. *__* Though I've heard they have some poseability problems.
~If I buy a doll, does it come with a wig/eyes, or is that a kit?
If you buy a "full set" you'll get wig/eyes/clothes. Most companies will send random eyes with the doll, and some might send a random wig, depending. I know Jeeryama on Ebay will send you a random wig if you buy a full doll through them. In general, though, you have to purchase everything separately.
~Any other general terms or whatever I should know?
The big dramatic one is "recast"--which is a bootleg doll made by a third party. They take legit dolls and make molds from them, then make cheaper reproductions and sell them for less. This is a HUGE point of contingency in the doll community, because counterfeit items are illegal to buy and sell, and buying and selling is a big part of the BJD hobby. (Not to mention it undercuts the companies that originally made the dolls and hurts their sales.)
Otherwise, though, I can't really think of anything! If you're not interested in joining DoA, there are some nice Facebook groups you could probably explore if you were more inclined toward that kind of community. c: