London's Heathrow Airport is an easy place to arrive. You don't need much of a plan - just follow the signs to the Heathrow Express, which takes you from the airport to Paddington Station. From there, you are on the Underground or in a taxi.
Heathrow Express runs every 15 minutes, with the first train from Heathrow to Paddington at 5.12am and the last at 11.48pm, and from Paddington to Heathrow from 5.10am to 11.25pm.
That's 150 trains a day and the trip takes 15 minutes.
In my experience, it was clean, comfortable and easy.
It costs £18 one-way ($28), return is £32 ($50) with the ticket lasting a month. The machine to buy the ticket is simple to use with a credit card or cash (British pounds), and there's an ATM next to it.
At Paddington Station, there are also airline check-in terminals for those heading out to the airport.