If you have a suitable contactless debit or credit card, the easiest would be to buy a return ticket from Gatwick to London, then use your contactless card for bus and tube travel. You can either opt for the expensive Gatwick Express service into London Victoria, or save a few quid and take the Southern service from Gatwick to Victoria. The Southern service is only a few minutes slower, but you'd have to wait on the platform for it, while the Gatwick express normally has a train waiting to board.
Using contactless, as with Oyster, there's a daily price cap applied. If your journeys that day go over the cost of a travelcard, the price will be capped at that. Using contactless, TFL count up all your uses, and make a single charge at the end of the day, so if your card has a per-transaction overseas fee you'd be charged this once. (If staying several days, using an Oyster card might be better on that front, as you could top it up once, vs the contactless total going out daily, but that all depends on your card's overseas fees)
With no contactless card, I'd normally suggest getting an Oyster card, loading it up with 15 to 20 quid, then using that within London.
If you want an easy time, and don't mind paying more, when you get to Gatwick buy a Gatwick to London with One Day Travelcard for your day going in, and your day coming back. That will cover you for the train journeys, and all buses / tubes / trams / etc in London on those days. Costs more, but is easy. Get it from a ticket office, as the machines generally don't sell tickets for future days. Probably an off-peak one will do you, but ask. Ticket prices and restrictions listed here, but there are lots so best ask in the ticket office. You're looking at between £15 and £40 per day, depending on trains taken and time used. For the extra days, easiest (assuming no oyster or contactless) would to also buy a Clapham Junction to Victoria One Day Travelcard from the ticket office. Same price as buying a One Day Travelcard paper from TFL, but you can get it all at the same time