Planning a first-time week-ish long trip to London/Southern England in May, questions about day trips, best general recommendation, hiking?

Some time around early/mid May, I'm adding London to a longer western Europe trip as my flight back home goes through London anyway. I can spend probably up to a week in the area. The itinerary is wide open.

I grew up in a city but now live pretty rural, and want to experience both in this trip. I'm an avid hiker and really enjoying getting out of society and exploring wherever I am. I'd also love to make it to a coastline, but have read so much about so many places that I'm getting analysis paralysis.

For the London portion, where are the best areas to stay, for walkability and bikeability? Looking for hostels. I'm an Atlas Obscura type person and love the weird and unique sights and spots. I thoroughly enjoy biking, live music, and parks. I don't drink often at all and won't be focusing on it while I'm there. Where are the more authentic/locals parts of the city?

If you had a week in the region, how many days would you spend in London, and how many day trips out of the area? Any towns/destinations accessible by metro out of the city worth spending a night or two in to gain deeper experience?

Please share your wisdom. I'm open to any advice! Thanks so much.