I too would second getting added to bills ASAP (even just council tax or water bill or something).
RE establishing credit, it might be difficult to do right away so I wouldn't worry about that too much. DO look at sorting out your NIN. It just gets it out of the way and is one less thing to think about.
DEFINITELY save pieces of your mail/joint mail that qualify for usage on your next visa. I wish I had known that as it just made it that much more daunting when we went to renew vs if we had looked ahead and stayed organised from the start (start a folder you can keep filed away).
On a slightly different theme to the above, I honestly wished that I had put myself out there more and established a positive routine. The first month I was here, before I got a job but my husband was working, I would literally sit around the house and unpack and watch TV. I would go out to grab a coffee and walk around the shops and come back to make dinner. It was a bit lonely and isolating but it was totally within my power to change as I was in a big city with lots going on. I had somewhat of a routine but I wish I had done more to try to make my own friends as opposed to relying on my husband's friends. I wish I had joined classes or even a gym to get in the habit of getting out and doing something healthy/active for even just an hour. I should have attended meetups and done all that as it just got harder to put myself out there the more settled and comfortable I got with my new surroundings.
Also, if you have the opportunity, I would maybe practise driving if you have a US license and access to a car in the UK as you can use that for the first year before needing to get a new provisional license. I should have got all the practise I could get in during that first year but where I was living and the one car that we had being above my experience level, I just wasn't interested and I put it off.