When I was at UTI I decided to get tools rather than the box. Look for a used Snap-on, Mac or Matco tool boxes or even Craftsman. The tools are what help you make the money. Worry about the tool box once you are making the cash. Get good sets of sockets, wrenches and ratchets, and basic hand tools. Once you get in to the shop enviroment then worry about special tools.
Good idea. A clean used box from off a local truck will suffice to get you started.
Three other considerations:
1. As mentioned earlier, know what you want to work on. When I worked dealer 1/4in - 1/2 drive were important, sockets & wrenches above 3/4in or 18mm were rare. Transitioning to bus rarely use 1/4 inch drive, never used metric so all that investment went home, had to purchase a bunch of specialty sockets, complete 3/4" drive set and wrenches and sockets all they way to 1-1/2". 1-1/8" was the most popular size. Buy impacts over chrome. Impact sockets will work with hand ratchets but using a chrome socket or extension on an impact gun is not a good idea.
2. Work rules if at a Union shop and some non-union shops have provisions to buy larger and very specialized tools. That should also be a consideration. When starting out in this career Do Not Purchase what will be provided by your employer. At both the dealer & bus shop everything over a certain size was purchased by the company. I finally got tired of tracking down this socket or that wrench and purchased my own, but by that time had the basics covered. Also Snap-On sells 'too cool for school' looking safety gleasses. Really? Who you trying to impress? Save your hard earned money and just use the workplace issued Birth Control ones. The blonde model in the convertible is not going to give you the time of day, she wants a ball player.
3. When new in the shop it is natural and kind of expected that 'the new guy' will not have everything he needs. Borrowing a tool once, maybe twice is allowable, but if you need to ask a third time...don't. Call Snap-On and purchase ASAP, don't need to get the reputation as a needy flake.
About UTI...when I interview potential mechanics that have survived the crueling testing process that human resources put them through UTI certificate holds more weight with me than being ASE certified. I have only had one UTI grad not work out in my 15 years of supervising.
Good luck.