As ptiger says, there are fellowships that you can compete for once you are in your graduate phase. There are two types of money, internal (from your institution) and external (from the NIH, private sources, etc.).
You will not actually see this money yourself, but it is applied to you in other ways. Depending on the institution, you may receive a kickback of a couple thousand dollars per year if you successfully apply for funding. This way, the school promotes its students to get that fudning. Ask about the school's policy when you interview there. The rest of the money will go to your lab and your institution. Many intramural sources of funding go towards your education, and will pay for things like conference travel or seminars.
If anyone knows specific places MSTPs can apply for this, I would be interested to know. I am going straight from what MSTP students have told me, but I have never found specific sources of funding myself.