You will need at least a set of basic hand tools to begin with. Hopefully your shop has a required/recommended tool list for you. My recommendation is that you don't buy anything but the necessities until you have at least a month or two on the job to query coworkers and witness firsthand what works and what doesn't. I have seen fellow A&P students make a ridiculous number of terrible purchases from the tool truck just because they are afraid of losing the discount after they leave school. If you absolutely MUST buy tools before you are out of school, DO YOUR HOMEWORK. It will save you hundreds, if not thousands of dollars. Talk to your teachers and see what they recommend, google search forums, etc. Your teachers will probably be some of your best resources. I have yet to meet one who doesn't like to talk about his tool collection.
As you start your job, you will find that some tools are really only worth buying if they are Mac, Snap-on, Matco, etc, while others can be Harbor Freight, Craftsman, Kobalt, etc, and it will make no difference at all, except in your own pocketbook. If you are really into status symbols, go ahead and buy all Snap-on or whatever, but if you like to keep your money, DO YOUR HOMEWORK! Even if you feel like you need to buy brand-name for a particular tool, Ebay and Craigslist can be your best friends, especially if you are looking for a big ticket item like a torque wrench or a big set of something. One thing that I have noticed on Ebay is that Snap-on especially has the name recognition, and it is tough to find bargains, but many of the better tool makers such as Cornwell, Proto Professional, S&K, etc. do not get as much traffic, and it is easier to score deals. I bought a 7/16-1 1/4 combination wrench set that is Proto professional and every bit as good as Snap-on for $40 including shipping!! You do give up a little bit in tool truck service, but they are really good tools for a lot less than retail. Craigslist is a little bit more hit and miss, but again, it can be good for big ticket items. It takes a lot of patience to decide what you need, and then purchase it secondhand, but it will save you tons of money.
I have been a mechanic for a year and a half, and I still keep a post-it notepad in my drawer to write down tools that I want to buy. I would highly recommend this practice. The rule of thumb that you will see everywhere is that if you borrow something 2-3 times, you need to buy it. Hopefully you will be in a shop where your coworkers are patient enough to loan you tools while you slowly build up your collection. I cannot stress enough how foolish it is to get way in debt when you are beginning your career. Skip the pop and chips on break, drive a clunker, do what you have to do to start saving money so that you can buy your tools with cash. You will thank yourself in a couple of years. Ask for tools or gift cards for your birthday and Christmas presents. I am fortunate to have a VERY understanding wife who allowed me to register at Sears for a whole bunch of tools. I wound up with $400+ in tools and gift cards!! (Gotta marry right for that one!)
All that said, starting with the basics, I will try to walk through different categories of tools and make recommendations in each one.