Hi wanderer786. Welcome to FT.
First, I will agree with wcj that you cannot play the credit card game if you carry a balance, unless it is on a card with 0% APR (some cards offer intro 0% APR for the first 12 months, or the like). You should always plan to pay your credit card bills in full each month. Also, you should make sure that your school does in fact accept credit cards for tuition payments, and does not charge anything for doing so.
Otherwise, putting your tuition and other school costs on a credit card is a great way to make a lot of miles and points. I think what you are trying to do is definitely doable. As for choosing the right card, we don't have much information to offer you advice. Are you only interested in airline cards, or do you need hotel points as well? What is your home airport? (That gives us an idea of what airlines you are able to fly). Do you have any miles/points in any programs currently?
In addition to having good credit, you must also have a good credit history, otherwise it may not be very easy for you to get multiple new cards. For example, hypothetically, if you've only ever had one card with a $1000 limit, but always paid it in full, you would have good credit, but would have difficulty in getting more than 1 or 2 decent cards approved. I assume you don't have any cards that would be helpful towards your goal? (cards that earn points that can be transferred to programs).
Sign up bonuses are the best way to get large chunks of miles quickly. But if you have a large transaction, like $25k, that you can put on a credit card and pay back, that's a pretty easy large chunk of miles as well.