Hi there-
I'm moving to Madrid in January for about 6 months and am wondering what to do about health insurance. Does anyone have any recommendations? Also, if you have any other advice on what to bring or not bring, that would be very helpful. Thanks!

I'm not sure what are the prices for a tourist, but I think you should be able to get a complete six months-term health insurance for about 300 euros or even less (add about 60 euros to total price for basic dental assistance).

Anyway, in case of urgency, I think that government hospitals are forced by law to take care of you, even if you are not a Spain's citizen. They must take care even of immigrants with no residence permission, so.... If you are thinking just in preventing an urgency, I believe that you don't need a private health insurance.
I think that's the current legal status for foreigners health care in case of urgency. if it's not, I hope someone will correct me.

When we were in Spain for 6 months, we maintained our US coverage, which would have reimbursed us for care in Spain. We made a few dr. visits and paid cash. It wasn't very expensive, so I didn't bother to submit to our insurer. Our Spanish friends steered us to private clinics, not because of better care, but because they suggested it would be less hassle to pay cash there. So contact your your US insurer and find out what kind of coverage they offer when you are outside the US. If you don't have health insurance in the US, sorry I can't offer advice. (Our health system rots!)

If you don't have insurance in the U.S. and you just want to cover emergencies, you could try the blue cross "emergency" plan. It's as cheap as they get, has a high deductible, but will do if you find yourself in some big mess. The premium is around $50/month, depending on where you live now.
If you're a student and under 26 you can check the benefits from the ISIC card???