One of the nice things about bikes is that they're mechanical, with almost all of the moving parts in plain sight. It's relatively straightforward to understand and maintain them.
The commonest repair you'll need is fixing a flat tire. It's something I think every biker should be able to do. You'll need a patch kit, a couple of tire levers (one of mine is this one), and a bike pump. Modern mini-bike pumps are much smaller and more efficient than anything you're imagining if you haven't used one lately.
As I noted with headlights, these tools aren't useful if you don't have them with you, so make sure you make it easy to actually carry them. I recommend having a bag for your bike gear that's really easy to put on and take off. There are a lot of options here, from under-seat bags to "trunk bags" that mount on a rear-rack (without interfering with your panniers.)
I also like having a multi-tool that lets me tighten and adjust almost everything on my bike. A couple of times, my chain has broken while I was out, and the chain-breaking tool is what allowed me to keep going.
Many bike repairs require special tools. I've acquired a lot of these over the years, and do almost all my own repairs and maintenance. If getting greasy and fiddling with mechanical things isn't appealing, you may never want to go down that road, but you'll be a happier and better cyclist to be able to perform some simple fixes in the field.