I think its looks very good especially for someone new at this. The tracks under the STM32 are no problem. Your use of a ground plane is a good choice, the circuit will be pretty quiet but this isn't really necessary- lots of microcontroller boards like this were made double sided without any planes. You have to take some care with bypassing and routing grounds well with fat traces. I love the test points- nice job. With boards being so cheap and quick these days, I'd check it over one more time and send it out for fab. It will work even if there is a little rework. You'll probably have a final cleanup pass anyway for little things or parts availability, etc.
I noticed a few small things that are mostly just comments-
1. Those two USB traces going through R3 and R4 are routed awfully close together- as long as your board vendor can do the line widths and and the spaces, its no problem but I don't think there is a good technical reason for this. These are USB D- and D+ and while they have to be matched, etc, they don't require this level of effort.
2. All the traces on the board are pretty small, they look like 6 or 8 mil. If you're hand soldering, you might want to make all the traces 10 or 12 mils where they'll fit. Tiny traces are easy to overheat, lift pads, etc. Use the most conservative design rules that will work- its easier for everyone and improves yields.
3. I don't see any mounting holes, this can be easy to overlook when you get buried in the routing.