Am I best to get the train? Bus? And which ones?
"Best" in situation depends on what you value most - time or cost.
If you want to get there fastest, and money isn't a concern, then get the Heathrow Express train, which will take you to Paddington. This leaves every 15 minutes and gets you into central london very fast, but it does cost something like £25. From Paddington you'd then need to take the tube - using the District Line (the green one) heading towards Wimbledon. If you can't get one directly to Wimbledon from Paddington quickly, grab any southbound District Line tube and change at Earl's Court for a Wimbledon bound train and get off at Fulham Broadway.
If you don't mind taking a long time to get there and want to save money, you'd be best taking the Piccadily line from Heathrow - this tube line will take you to Earl's Court where - as above - you can catch a Wimbledon bound District Line train and get off at Fulham Broadway. almost 8 years ago
Picadilly line to earls court, then district line to fulham broadway was how i used to go, quicker and cheaper than cab. Get an Oyster card if your staying a week or save, you can use them on the trains buses and tube and saves you some $$ almost 8 years ago