Hi Judes,
I hope you have a great first trip (of many in the future :)
I've stayed in Hampstead twice, Southwark, Windsor, and Paddington. I'm a low-stress traveller and have done SO much planning, but it's still a VERY large place and things don't always go as planned. The Pax Lodge hostel in Hampstead is run by the Girl Guides/Girl Scouts, has a GOOD pub right around the corner, a dining hall, en suite options, grocery store down the hill, kitchen for residents, and buses that stop on the corner (and 2 Tube stops, one up the hill, the other down the hill). The buses are the 24 that goes right to Trafalgar Square, and the 168/268 to Waterloo. So if a local neighborhood, and Camden/King's Cross, Hampstead Heath, Kenwood House, British Library is on your radar, it's a welcoming option for the whole family.
Paddington's there for the Heathrow Express, lots of small boutique hotels near Hyde Park, Kensington, good buses. City neighborhood, tho.
Southwark was OK too, a bit of a challenge with getting connected to the sights on Sundays, and can be noisy in the Waterloo area, but an option. Near the river, near Burrough market, I've only been there on days it wasn't very open, so scheduling can be a challenge. Near the Globe Theatre. Good river walks.
I may venture to Greenwich in the future cuz I'm an Astronomy club member and have never been there, but it IS far, and if you have a list of places you would like to go, I would suggest you center on mapping out your preferences, and staying near the most of them that you can.
I've found that as a traveller, I need some basics in commercial hotels, charming varieties of places work for Rick Steves, but charming has never worked out for me. :) (long stories) I've finally accepted that I need to be comfy at the end of long days, so feel free to be yourself and do what you need for accommodation. August won't be cheap, unless maybe you stay at one of the many college dorms that are rented out, but useful accommodation when you get to London for the very first time is precisely why god invented credit cards :} Low-stress.
Travelodge and Premiere Inn have both been good (3 times), boutique hotel OK/fancy but drafty, Pax Lodge welcoming but I'm a Lifetime Girl Scout and volunteer for the World Foundation (& volunteered there with my daughter for 2 months) so I'm partial to it in some ways.
Basics suggestions for a new person, make sure you have data and phone service there. I have a Google Fi phone that is $23/month unlimited data, and worked within 5 min of touching down in London for the same price as home. HUGE difference from all the visits scrambling to try to buy phones over there.
Google Maps for sure, and some say CityPlanner apps, but I have not had success with it as much as others have.
The Victoria & Albert Museum, possibly a whole day or combined with Kensington Palace. Amazing place, I preferred their jewelry room over the Crown Jewels. Harrod's might have just been having an off-day when I was there, not real fun, cranky people, but maybe you'll have better luck.
The British Library! treasure room, and the St Pancras Hotel (tea?) near there and King's Cross.
I drastically prefer buses over the Tube, would rather SEE the place instead of being in a dark tunnel, and I have physical challenges with the high-speed STEEP escallators, find it MUCH easier to step on to a bus. My next trip will involve Discover Real London taxi tours, he donated a tour to Pax Lodge's volunteers, super nice guy. When everything's gone wrong, I drop my attachments to plans and get a taxi to help fix the problem.
I always try to find events at clubs or similar churches/temples that I can meet real people. Meetup.com has loads of options.
Love to stay in Windsor, super close to Heathrow (it's VERY far from the enter of London), there's a trick to the travelodge and the trains, but once we sorted it all, it was good.
Good luck! -Alison