In 15 years I've never needed more than a screwdriver with an extendable magnet built in to pickup screws that have tried to escape lol and a philips bit to undo the screws. Oh and a pair of cutters to snip the cable ties OEM machines often have wrapped round the cables.
A multimeter can be useful but not essential, i carry spares to test components such as the PSU. I've never used a wrist strap, never will poxy things. If you want to buy some kit to look professional get a can of compressed air and some isopropyl alcohol. They will get used at some point unlike most of the tools in these PC tool kits.
I hope your college course is better than the BTEC national Diploma I did......what a waste of two years! I spent the first year word processing, doing spreadsheets etc and programming in pascal (pointless) didnt even see the inside of a PC till the second year, what a joke!