Matt,
I am a saxophone player that started a small computer-based home studio for my own use about five years ago. I am not a studio engineer (and I did not stay at a Holiday Inn Express last night), but I have picked up knowledge along the way that has helped me. While I no longer consider myself a "newbie", I probably am still a novice compared to people who REALLY know what they are doing!
First off I recommend a large diaphragm condenser mic for all saxophones. A large diaphragm condenser mic will give you a more true representation of a true saxophone sound. Dynamic mics like the SM57 are very good for live performance, but will not allow the expression and full body sound that a large diaphragm will give you. I use a Studio Projects C1 large diaphragm condenser mic. It sells for $238 at Sweetwater.
Since you are a student, I am not un-sympathetic to what is probably a limited budget. If you cannot afford the gear, there are other things you can do. You can use what you have and make the best of it or find someone who will allow you to use the gear. If you are at school, do you know someone in the music department on the recording crew or someone in the recording engineering department. Perhaps they could trade time or favors.
At any rate, here a some basic techniques that I have used with some success. First of all you need to decide what type of sound you are looking for? Do you want an old school jazz sound like Paul Desmond, John Coltrane or Ben Webster? Or do you want a more modern sound like Dave Koz or David Sanborn? Perhaps you want to play rock and roll like Mark Rivera on many Billy Joel recordings. This is the question you must answer first.
Once you have that straight, here is what you can do. For a darker sound mic to the side of the bell, not directly towards the bell. Have the mic 6 - 9 inches away (but to the side of the bell. The C1 has a cardioid pattern which means that it is somewhat directional and works well in this instance. I like this setup for a lot of my recordings, especially ones where I am trying to blend with myself (recording a saxophone ensemble by myself) or other people.
For a brighter sound, you can move the mic in front of the bell, but pull the mic back to around 12 - 18 inches. You do not want to be too close or the lower notes will really overdrive the mic. The further way the mic is too the horn, the more room ambiance you will get. The sound is less "detailed" the further away you get as well. How well it will sound depends on the room. A condenser mic gives you that flexibility. With a dynamic mic like an SM57, you really cannot get too far away reasonable results.
Some people use multiple mics (one on the stack and one near the bell) and mix them together. I think that is overkill, but it would capture more detail in different ranges of the horn.
Effects are often overlooked when evaluating a mic. This is a mistake. A saxophone completely dry is not very pleasant. Of course when you record a track, record it dry, but in the mixing phase (or just when listening) effects are crucial to getting a good recording. But as many here know, they can be and often are over-used, so it is a fine line between them enhancing a recording or making a recording over-produced or "artificial".
Start with compression. Compression is necessary when recording winds because dynamic range is very wide. Some suggest using compression while recording in order to get a consistent level without clipping. Use a small amount of reverb, but not too much. The old adage with reverb is add it until you can hear it and then back it off slowly until right at the point where you don't notice it. That makes it just about right. EQ is important, but as with anything, do not over use it. Back off frequencies below 80 (or even 100) to avoid "thumps" caused by moving feet, chair legs, mic stands and also key noise. You may also slightly back down on the high end for key noise as well, but not so much you loose detail and overtones.
Well, these are my basic suggestions with the knowledge I have gained by reading articles and my experiences in my studio. I am still learning and will continue to as long as I am doing it. In the end, just record, record, record! Experiment with different mics, mic placements, effects and if possible different rooms. Even on a limited budget, you will eventually hit upon a combination that works for you.
I hope this has been helpful.
Joe