Both the UK and USA allow dual nationality, so there is no need to relinquish US citizenship if you don't want to - it might prove advantageous in future.
There are minor inconveniences - ask Biris Johnson!!
The USA does tax worldwide, so you do have a requirement to submit tax returns, but there is a reciprocal tax regime so as you pay your UK taxes it is just a paperwork issue, and not overly expensive - there are a lot of accountants who can do it for a few pounds.
Boris found they would not let him in with a UK passport, requiring his "national" US passport - so If you haven't got one, get one if you wish to visit.