I'd say definitely travel by PAYG oyster, there's no drawbacks to it, it's the cheapest method, and you get your card deposit back if you return it when you leave (has to be at an underground station - not an overground or train station).
It'll probably be nice weather but literally the weather could do anything
so I'd say bring clothes for warm weather, but be prepared with a jumper and/or a big coat as well just incase.
In terms of the area you should stay, I don't think it matters so much as long as you're close to a tube station. The transport's so good that if you're in Zones 1 or 2 and close to a station, it shouldn't take you too long to get anywhere in and around central London.
Must-go places just depends on what you like, there's so much you could do. One of the first places I visited when I moved here was the Tate Modern and it's still my favourite experience of London
But if you're not into stuff like that you'd hate it. If you like the theatre then I'd recommend 'The Book of Mormon' which is a satirical musical, or 'The Play That Goes Wrong' which is a comedy play.
I'd also recommend visiting Camden Market. It has some great vintage shops and is really popular with tourists (be VERY careful with your belongings though because you get a lot of pickpockets round there), and it's a short walk to the Amy Winehouse statue
And the parks in London are great. Hyde Park and Regents Park are my favourites. But like I said, it just depends what you're into really
