Hello... I’m from Edinburgh and come down as often as I can. (Down today for the Community Shield at Wembley). We got the train from Edin Waverley - Kings Cross. Usually we’d fly into Heathrow or London City airport.
As already said, you want to use the tube and get to Fulham Broadway which is right beside the stadium. (If you have a contactless card just use that on the tube, it’s the cheapest way to travel and will accumulate your usage and charge you accordingly) - Me and my brother have been everywhere today on the tube and been charged £6.80 for the days usage.
Enjoy the game!