Agree with Marco on the Bishop’s palace (it’s amazing) and North End Road, which we love for food shopping and people watching. Just before the Fulham Broadway bus stop at the end of North End Road is The Cock Tavern, which is a friendly local non-touristy joint.
We have not been yet, but Fulham Pier has been advertising a lot and looks like fun: food, market stall stuff, etc: https://www.fulhampier.com/
Stamford Bridge Stadium, home of Chelsea FC, is a draw for football fans, and just a 2 minute walk from the Fulham Broadway tube stop. There are tours, if you are into it.
And you’ll be a short bus ride from Kensington, South Kensington, etc., for more upscale/trendy stuff, museums… depending on exact location in Fulham it’s quite easily commutable.