Basics are where it is at when you're starting out, regardless of the type of work. Nobody expects you to have everything, but they also don't want to loan you basic, commonly used tools. You want to own:
Excellent set of screwdrivers. #2 phillips is the most used tool on any job. #1 phillips is not far behind. Lots of flat slot screws also on trim and hose clamps, etc.
Larger flat screwdrivers to pry with when needed, in addition to pry bars.
Some kind of 1/4" hex bit driver with wide assortment of torx and hex bits for it. This gives you some good universal capability. Lots of lights and trim held on by torx or hex now, and lots of hex cap screws to remove. I'd buy a Williams or Snap-on ratcheting screwdriver. It will become one of your most loved tools. While you're at it, buy a 1/4" drive, hex bit driver socket to let you use your 1/4" ratchet/breaker bar with hex bits. Really, really handy sometimes and better than a 1/4" socket because the bits stay in it.
Good set of small chrome finish sockets, either top quality 1/4" drive or good quality 3/8" drive. As said before, the larger stuff is less used/easier to borrow/shop supplied. You'll soon know if a 1/2" drive set is needed. What they will expect you to have is a small set to take off brackets, accessories, electrical connections, etc. etc. etc. These are your workhorse tools for most common tasks. Buy as wide a range of lengths of extensions as you can afford. You'll need a lot sometimes.
Good set of wrenches in the 1" minus size, and 24mm minus size. Again, universally needed, these are your workhorse tools.
Wide range of high quality pliers. Absolute essentials; workhorse tools. Almost never complete any task without using a pair or two.
Wire brush, picks, putty knife/scraper, utility knife. Always cleaning stuff off BEFORE you work on it and have the crud fall in your eyes.
Mini maglite. Don't use the flashlight app on your phone.
Hammers: small ball peen, dead blows or soft face, 4 lb sledge, 2 pound ball peen. You'll soon see what others you need.
Breaker bars. One in 3/8, one in 1/4, and a top quality, long one in 1/2". The leverage will save you from slipping and hurting your hands. Even with air tools, there are things that need breaker bars.
Inspection mirrors.
Magnetic and 4 finger pickup tools. I have to use those a lot more than I will admit, and they really save you sometimes.
Good quality leather gloves to protect your hand when you carry heavy stuff and when you have to push and pry.
Nitrile gloves to protect against grease and oils. Shop should supply those.