A small first aid kit is one of the most useful and important parts of any reasonable every day carry kit.
For minor cuts and such, you don’t need much more than a few bandaids some gauze and tape. In general, the small cheap mini first aid kits are a good starter point. Johnson & Johnson First aid to Go costs a bit over 2 USD, and the case is compact but solid with some extra space.
What I suggest is getting one of these and adding a bit more so as to turn it into a more well-rounded kit. Include a couple more alcohol swabs, some ibuprofen, aspirin and diarrhea pills.