I've done some trips that have made my mother TERRIFIED, but regular emails tend to set parents at ease. I think if I had gone for a trip to London during university, she'd be far more excited than the trips I did go on. Just stay in regular contact with her to let her know you're okay and enjoying yourself. There are internet cafes everywhere, so a mobile phone won't be necessary, but you could look into getting an unlocked phone in the US and buying a UK SIM card once you arrive. My parents have done that every time they're in the UK and it's an easy way to stay in touch.
London is a safe city as long as you're cautious like you would be anywhere else, as everyone else has said. I've walked around the city and travelled alone on the tube at night and have felt completely safe. Staying at hostels is a good idea, as you'll meet fellow travellers around your age who are in the same position as you. Petty theft is the only major issue you should encounter, including at hostels, but just be sure to protect your belongings. There's no need to wear one of those ridiculous body wallets under your shirt (something like that is more likely to paint you as a tourist and attract a mugger, IMO), but obviously do be aware of your bag/wallet when you're in public and especially on the tube.
You'll definitely want to interact with "the locals", but as with anywhere, don't be too trusting of anyone. Don't get too drunk when you're out alone and make sure you're staying in a safe neighbourhood. Guide books are good at discussing the safety of neighbourhoods. They're also great for helping you get out of the typical tourist areas, and exploring the best parts of the city. London's tourist areas tend to be a bit annoying (I can't stand crowds!), but once you escape them, it's a fantastic city.
As for being confused and possibly getting lost - you probably will be in London, but the transport system is great. When I've been lost, I've simply walked a bit further until I find a tube station or bus stop, and can then figure out where I am. I always take my iPhone/iPod with me when I travel and use the wireless on it (turn of mobile data, though!) for the maps. You can load directions on Google Maps when you have wireless, then use the map when you're walking around and no longer have access to wireless.
Definitely try to save up as much money as possible, as the UK, particularly London, is expensive with the exchange rate. As long as you plan ahead, it's very doable on a budget, though. You can get a really good, cheap lunch from places like M&S (department store with great food markets), take public transport for sight-seeing (skipping the £50 tourist buses), and go to the museums, which are free, instead of expensive tourist sites (Westminster Abbey, St Paul's, etc). Also, avoid London this summer, especially in August, as prices will sky-rocket due to the Olympics.
Hope you have a fantastic time!
