It may depend on where it is you're headed once you get into the city. It might even be a better idea to fly into or out of Gatwick for some destinations, such as London Victoria.
Realistically, the most stress-free way to travel into or out of the city is to relax in the back seat of a chauffeured car service. Although I recognize that this isn't an option for all travellers.
Navigating your own journey, the most convenient route might be via the Heathrow Express train service. It's a quick train journey departing Terminals 2, 3, and 4 - with another quick transfer from Terminal 5 available for certain international arrivals.
However, the convenient non-stop service of the Express means that it only services a single station, London Paddington, which is located on the north side of Hyde Park. If you're staying toward Covent Garden or any of London's hip East End you may find it less convenient to connect via the Underground at that point.
I often recommend my travelling clientele stay in London's Piccadilly, Covent Garden, or even Holborn areas in order to make the most out of their visit to Europe's largest city. From these neighbourhoods you can easily walk to most of the historic landmarks or at least dive underground onto the sprawling Tube system.
Departing Heathrow for the city means a somewhat more straightforward service on the Piccadilly Line, where a transfer won't be necessary to the abovementioned stations. Riders can simply enjoy a 45-55 minute direct route to the centre.
My single caveat for taking the Underground is that your stress level will increase with the size and number of bags you're carrying with you.