London is a city set up well for walking and public transportation–I simply see no point in having a car here and would never want to fight that traffic on my own. Whenever you need to plan how to get from A to B via train or bus (and would like to gauge the amount of time to allow for this travel), we recommend that you bookmark the Transport for London website: https://www.tfl.gov.uk
Here, you can find route maps, timetables, how to buy tickets (including procuring your Oyster card) or use the Journey Planner feature to enter your starting and destination locations. The site also posts live service updates or, for your added convenience, you can consult the Travel Tools page to sign up for mobile travel alerts or email notifications of weekend closures–there is always one line or another that is closed for scheduled engineering works over the weekend, which definitely cramps one’s plans when you show up at the station expecting service to run the same as it does any given weekday.
London Relocation Ltd. has your flat-searching transportation covered, as we hire a driver for the entire day–but when you’re officially on your own, TFL will be a necessary resource. Happy travels!