I feel for you as I've had a noisy adjoining neighbour for several months and polite requests have failed. Well, not 100% failed, neighbour apologised, said they hadn't realised (really!?), offered to be more careful.... hasn't really improved. Very odd as, on paper, the previous neighbour should have been noisier, but while I was aware of their presence due to some sound travelling, it didn't affect my household's ability to enjoy our home, relax or sleep. Right, back to your situation...
I recommend keeping a noise diary. I'd write down the date, start and end time, type and rough volume of noise and (very importantly) what effect it had on your life.
Best case scenario: it reveals that the frequency/impact of bad events is less than it feels, especially if these really are noises related to the early days of moving in.
Medium case scenario: you can choose 3 or 4 of the worst events to mention to your neighbour when you next share your concerns. Don't share all the minor events, if neighbours don't improve for the 3 or 4 major issues, they'll not improve for all the others and a list of 25 issues probably won't go down well.
Worst case scenario: you have a record to share with Environmental Health (or letting agent, or landlord) if you ever go down that route.
When you next speak with your neighbours, consider taking the approach of "we all need to be able to enjoy our homes, when we hear X, it causes problem Y, can you help us fix that? We won't be perfect and the walls are pretty thin, also we enjoy being in the garden in the summer, what can we do to help you enjoy your home?" Write a summary of the conversation (both sides) and store it with your noise diary. Relevant as playing / barbecuing in the garden can be noisy, and windows tend to be open in summer.
OK, these things work if your neighbours are considerate and reasonable. Unfortunately, many aren't.
Good luck.