Check out the YHA hostel near St. Paul's Cathedral, for you first visit:
Great location, 35 pounds for a private room. This neighborhood is right in the midst of sights you'll want to see: 1. St. Paul's Cathedral 2. Tower Of London, to name two. Easy to get around on the tube (underground). You can get to other sights using the tube. You will want to see 3. Westminster Abbey(full tour), 4.Big Ben and Parliament Buildings (a walk-by will do), and 5.Buckingham Palace (a walk-by or drive-by will do), 6. The British Museum (enough cool stuff on exhibit to keep your eyes popping for three full days). That's just a start. There's a lot to see in London!
The best way to get a glance at all these and more, for someone who has a short period of time in London, is the Hop On Hop Off Bus. It's a bus you can catch at any tourist attraction. You pay, get on; if you stay on the bus for two hours, you'll get the whole tour around London, with a guide telling about the sights you're passing. Your ticket is good for 24 hours from time of purchase. So you can get off the bus anytime you want, enter a sight you want to see, come back out, get back on the bus, continue on until you see another sight you wish to explore.
This is a good way for a first time London visitor to get a quick overview of all the sights, and decide which ones look interesting enough to "Hop Off Hop Back On" the bus. It also comes with a free river cruise past some of the landmarks on the river. http://www.theoriginaltour.com/
Also, just walking is a great way to see some neighborhoods. London Walks is a company that offers guided walks for $7 to $12; www.walks.com.