No visa required for US citizens. In fact I have done this many times.
To get to the centre of London, the following options exist (in rough order of increasing price).
There is a bus service to nearby hotels at £3 for a single trip - this however can be cheaper than a taxi.
Piccadilly Line  - an underground rail line from the airport to stations throughout central London. There are two stations: one for Terminals 1/2/3, and one for Terminal 4. The journey takes around 50 minutes. £4 (single); Travelcards (zone 6) are valid. This option is much cheaper than the Heathrow Express, and can be faster if your destination is in the suburbs west of London; use the Journey Planner to pick the best route. It can however, become very crowded and unpleasant once the train gets near the central area ( - Still Best Option).
By coach - National Express runs a regular service to Victoria Bus Station (next to the railway station) for £15 return - journeys take about 40 mins to an hour.
Heathrow Connect  - a cheaper but slower regional rail link between the airport and London Paddington station. The service runs every 30 min and takes 25 min from terminals 1,2,3. £9.50 (single). Railcards valid only up to Hayes & Harlington, after which a supplement must be paid to reach Heathrow.
RailAir  - a regular bus service to Woking, a town slightly southwest of London. Woking has good rail connections - if travelling from Heathrow to southwest London, going via Woking is often faster and easier than going through central London.
Heathrow Express  - a high-speed rail link between the airport and London Paddington Station. The service runs every 15 minutes and takes 15 minutes from terminals 1,2,3 or 23 minutes from terminal 4. £13.50 (single, booked on-line); £26 (return); Statistically the most expensive railway journey in Britain (at over £1 per mile; Travelcards (see below) not valid.
By taxi, the journey from the airport to central London will take approximately one hour. £46.00-£50.00 (not great value; you may save time and money by taking the Heathrow Express to Paddington followed by a taxi to your destination)
By Mini Cab, journey times as for road / taxi travel, however prices are generally cheaper than the black cabs. The Mini Cab service is a licensed taxi service scheme and cars must be booked in advance.
By road, the airport is some 17 miles (27 km) west of central London - a large part of the journey can be made by means of the M4 motorway which can, however, be quite congested at peak periods
By the night bus route N9. One of the few options if you need to get in or out of there in the wee hours. Takes about an hour to Aldwych, and only runs every half hour from about midnight to 5am (when the tube and other connections have stopped). Standard bus fare applies (about £1).
By bicycle. Travelling to the center by bike may sound like madness, but it's a lot easier than it might seem and an attractive alternative for the adventurous traveller. To get out of the terminal123 area, take the cycle path in the service tunnel on the left hand side of the main road tunnel. The cycle path is meant for airport employees and not widely advertised. It can be accessed by following the signs to car park 1A and turning right just before the car park entrance. After leaving the tunnel, turn right and head to Harlington and further on to Hayes&Harlignton station. From there, the remaining 20km journey to Paddington (in central London) is a pleasant, traffic free ride along the towpath of the Grand Union Canal. It will take you about 1h30min, which isn't all that slower than the Tube and it's about the only way to get to Heathrow for free. Most airlines take bikes for a small surcharge, provided you declare them as "sports equipment".