use public transport
you can get a tourist oyster which'll keep you going on the tube and if not you can always get paper travelcards which work on both the tube and on busses
never use taxis they're an utter scam unless you like the novelty of riding around in one (it's like new york, in that sense)
police are everywhere in london, especially in tourist heavy areas like oxford street and in central so i wouldn't feel to worried about travelling.
if you're going to places in north london, like camden and such (another great place i recommend you visit for the market and the canal there's plenty of street stalls which sell amazing jamaican food) keep an eye on your belongings and don't openly flaunt things. most tourist areas are safe though, and even camden has a strong police presence.
don't be put off by people saying "you're gonna get mugged stay in tourist areas" because the chances of it actually happening are quite slim, and major crimes like theft and stabbings are done in council estates or the REALLY downtrodden areas. go out and explore the lesser known areas of the city, it's a wonderful place with lots of historic value and culture.