I want to start learning the violin, but I don't know what's a good price for one. Should I get a used one or a new one? How can I tell that a violin is of good quality and worth the price asking for? What's a good brand?
I'm about at the same point (maybe a little bit ahead, but not much), so I'll help with what I can.
Violins are a bear to learn, for three reasons:
Because of these things, picking up the violin is not something to be taken lightly, and there is great chance of giving up before you climb the learning curve. So, my suggestion is to get an inexpensive instrument.
There's no big names for violins like there are for guitars. I mean, there are bigger names, but there isn't a Fender or a Gibson or a Martin. This makes it easy to skirt the no-shopping-advice dictum. On the acoustic side, violins are pretty much of the same material and of the same shape, so the good ones and the bad ones will superficially be alike. Beyond that, remember that the violins that were used by great classical players were not the ones that ended up in the hands of folk players, so, if your desire is to play fiddle, rather than classical violin, your desire is for an instrument that has different characteristics than a classical violin. Bows are a big determiner of the tonal characteristics, and so your bow could end up as expensive as your violin.
So, after that info dump, my suggestion is to find a cheap instrument and work with it. You should be able to find something between $100-$200, which is at the low end. (Fine contemporary maker violins go into the tens of thousands of dollars. I wish I could tell you what distinguishes them, but I've never touched one.) You'll eventually learn something about what makes an instrument sound good, but believe me, you need to work a lot with your ears and hands before the instrument itself is the determining factor.
There are only two factors, although they are broad:
These are difficult for a beginner to know. You haven't the experience to know if an instrument feels right. You haven't the skill to bow a note well enough to gauge its tone.
Fortunately, unlike a guitar, where fat necks and bad fret intonation can really hold you back, within reason, you can learn on a cheap violin.
So, if you can, take an experienced violinist shopping with you, let them play the instrument. They will be able to tell you if it feels right, and both of you will be able to make judgements about how it sounds.
One further objective factor, is whether it stays in tune. That's easy enough -- just check that the tuning pegs are firmly fitted.
If you can't find a violinist to help you, don't worry too much - consider learning with a borrowed instrument, or a cheap instrument, until you've learned to sound a pleasant sounding note. By then you should be able to tell a pleasant feeling instrument when you play it.
The best way to start learning is with a teacher. See if your teacher can help you find an instrument.
I agree with VarLogRant, there are no big names (in reasonable price range). Traditionally violins involve a lot of manual work, which limits the number of instruments by one maker. There are increasingly mass-market instruments more industrially manufactured from Asia and much of discussion is going on, how carefully the wood was dried and how likely they develop cracks later. They may also be good, but beyond holding the tune the criteria are not easily recognized. In any case you should take some lessons and the teacher should try the instrument.