It is so easy to venture away from the tourist areas in England, because no matter how far off the "beaten track" you go, you'll still be able to converse with everybody and read all the signs. The smart-aleck answer to your question of "how do you go beyond the typical tourist sites?" is, you just go. Find a neighborhood that looks interesting and go explore.
For more specific recommendations, it would help if you told us a little about what kinds of things you like to do. Do you like history, architecture, food, drink, natural beauty, music, sports, big cities, small towns, shopping, art, religious heritage...?
Outside of London, I'd suggest getting away from places like Bath and Oxford that appear on so many people's itineraries, and strike out somewhere completely different. Bristol, mentioned earlier, is a great city, just up the road from Bath, and visited by fairly few tourists. (I know it well because I used to live there.) It's full of charming neighborhoods, parks, and little museums - I particularly like the Georgian House and the Red Lodge, both listed at https://www.bristolmuseums.org.uk/. But maybe they're not to your taste, and that's OK - that's just one of many possible directions you could go.
Have you asked your family members about their favorite places to go and things to do for fun? They know you better than we do, so they might be able to come up with suggestions better tailored to your interests.