Re shopping: what type of shopping do you fancy -trendy or hipster? Mainstream or market? Different districts are massively different to each other. To get the best think about this carefully.
Re local cuisine: do you mean English English by this? The most popular dish was fish and chips and I always recommend this to overseas visitors BUT our most popular dish now is now an 'Indian' curry. Indian is a bit misleading because it can be Pakistani or Bangladeshi in influence but it's distinctly British and different from traditional authentic dishes from those countries. I also think that there's some great food to be found in Chinatown. Again it's 'British' in the sense that oriental dishes found in other places are different to British fayre. You also want to check out a British boozer - especially if you do fancy a drink - they are often good for 'traditional' English style food. If you do like beer try and find a good craft beer place as although they are less traditional they serve better beer and are culturally particularly friendly.
Re museums. There's lots. Go through a list and choose. There's billions of them. Most of the big ones now, ask you to book an entry slot online even if they are free on entry. Think about this carefully. They can get busy - you want to book the earliest slot available for the one you want to visit most. The British Museum and the Natural History Museum are the ones you should definitely consider this for. They are exceptional and it's easy to spend most of the day in either without trying. The Tower of London is great but expensive. Honestly given what's available for free in London, it's not necessarily the best option. You can see it from the outside from some distance as part of a sightseeing tour.
Re theatre - 100% the booth in covent garden recommended up thread.
The art galleries are also well worth a visit. Again most are free. But again plan to do early in the day and then another activity in the afternoon if you fancy them.
And as much as I'm loath to suggest it, a sight seeing bus / river trip for a day isn't the worst option for finding your bearings and working out what you really want to do.
Don't do the London eye. It's expensive. A good alternative which gives you great views of the city is the IFS Cloud Cable car. It's a fraction of the price and you can do it as part of visiting Greenwich.
Plan, plan, plan. Your homework will pay off with London. It is a VAST city. There really are not many cities in the world with so much to do. And whilst it is an expensive city there's so much to do which is for free, I'd encourage you to think about this as you will still have an amazing time and have more money to spend on eating out.
I'm not from London. But when I do go to London I make a point of making it count and getting the most out of my trips as a visitor. I've been many times and I've not done half the things I want.
Re trips out of London. Oxford is the classic for tourists. Honestly... There's other places that are better. If you can go more rural / further out do so. I find Oxford over rated for what it is. It's not representative of the UK either.
Trust me, you will get overloaded on this thread with the possibilities. So really think about what you want to do and why.
Tube is easy but very busy at rush hour. If you can walk during rush hour it's not necessarily a bad option. Bring comfortable shoes!