There's something that kind of confuses me when it comes to playing. My teacher sometimes uses open strings, but other times uses a 4th finger (prinmarily on the top E string and top A string). Why is this?
I feel like I'm beginning to understand, but would like as much input as possible (as I don't see my teacher often).
There are at least a couple of possible reasons. With the violin, and particularly the E string, using fourth finger on the string below to play the same note tends to sound better and the tone richer. The open E string can sound quite shrill. That said, it is usually easier to use an open string in a series of notes if you are going on to higher notes on the same string. Obviously, these are simply practices and not hard and fast rules, but I think many players tend to follow them. There may be other reasons I am forgetting, but those are probably the two main ones.
In the early years when I was being taught I was told to play my scales open strings going up and 4th finger going down.
When playing Baroque it is acceptable to use open strings a lot of the time, particularly the E if it is gut. A gut E does not have the shrillness of its steel brother but has a sound more in keeping with the gut A next to it, and, imo, gives a better sound from the 7th position upwards than most steel Es. Furthermore, when playing baroque music you don't use vibrato much, and when you do it will be as an ornament, usually in a slow movement. A lack of vibrato means that an open gut E fits in better.
Jake, when you have to play a fast passage in first position, make use of open strings to your benefit (if it makes those passages easier to play, and I would say, regardless of whether the passage is ascending or descending). For slower notes, the sound of the open strings comes into consideration. But, as Andrew notes, you can make an open string sound beautiful too. So, there are no strict rules, but a lot of common sense and musical options.
Sounds like you're a beginner/intermediate player -- so just do what your teacher says. :) Your goal is to develop a robust 4th finger so you might as well use it a lot.