I’ve started traveling really light. The folks I travel with have a pretty stocked trailer and so the kit I bring for myself has been pared down to just the essentials and I think makes a good starter kit for someone getting into the sport.
10 and 13mm ratcheting wrenches
T-Handle allen wrenches, 4mm-6mm sizes.
Ratchet with spark plug socket, extension, and hex ends for wheel nuts and stub axle nuts. Pick the sizes you need for your kart’s setup.
Needle nose pliers
Chain breaker for your chain size, probably 219.
If your track doesn’t have a compressor, then get yourself a small portable air tank and fill at gas station.
Pretty much everything else you can borrow, and if you find yourself borrowing, then buy it and throw it in the box. I almost never use sockets any more for anything thanks to the invention of the ratcheting wrench.
If/when it comes time that you want to change your own tires, then you’ll need a compressor, bead breaker, and a tool like the tire whiz or tongs. I got away with my first couple changes thanks to helpful folks who helped me through the first few changes on their equipment, then I bought my own. Eventually you’re going to want to get those kinds of tools, but you don’t need them day one. Chain aligners, chassis aligners (snipers), etc all fall into that category. You don’t need to be in a rush to get everything.
I have a whole garage full of sockets, air tools, etc and most of it never gets used.