Visiting England in March. Where to go besides London?

My wife and I are visiting England in March because my wife was born there and she hasn't been back in over 20 years since she was 13. I have never been before. We will be spending 2 weeks there. The 2nd week of our trip is dedicated to seeing and catching up with family mostly around the Liverpool area but the 1st week or so is totally up to us.

We want to spend some time in London and have allotted 4 nights in London. We want to see as many of the mainstream touristy things as possible for my first trip as well as a tour of Emirates Stadium as I am a big Arsenal fan (unfortunately their home game while in London is a London Derby against Chelsea and I've been told it will be nearly impossible to get tickets). I would like to see some sort of football match even if it is a lower league team. Are there any specific teams around the London area you would recommend that I could actually get tickets for? Also any recommendations for a London first timer outside the main touristy things?

Before we see family in Liverpool we plan on spending a night or two in Bath (with a trip to stonehenge thrown in there) and a night or two Cardiff for no other reason than we think that they look like cool places to visit. Pro and Cons to also visiting these places? Would you recommend alternatives to either of those places? Thanks for any info!