If you like maths then project euler problems can be quite fun.
I'd also recommend https://codecrafters.io/ - I did the Redis challenge recently while trying to (re)learn Java and it was a good experience.
I was in a similar position to you a couple of weeks ago - switched job after the better part of a decade. The main things I've found that have built trust are:
1. hanging out casually with teammates in the office
2. taking notes and/or screenshots in every conversation and then reviewing them to stay on top of everything
From the other side, I've onboarded a lot of engineers in the past and the best ones have been "yes, and?" people. So you'd explain a process to them, they'd say yes and try to figure out what comes next. e.g. if someone explains that changes are released via a pipeline, the best on-boarders I've seen would say "yes, and what stages are in the pipeline and how do they progress?". Where possible I'd recommend the pro-active questioning approach.