Staying at Bankside Autograph Marriott

Taking the Elizabeth line from Heathrow to Farringdon and changing for a Thameslink train south to Blackfriars will be quicker and a lot cheaper than any road based transfer. Or you could take the Piccadilly line from Heathrow to Hammersmith and the a District line train to Blackfriars and cross through the station exit on the south bank very close to your hotel.

The change at Farringdon has lifts and the change at Hammersmith is just literally a few yards cross platform. The Piccadilly/District route is cheaper, the Elizabeth/Thameslink route a bit dearer but more comfortable.

Neither route is at all complicated.

'"he subway (called underground here) and trains seem to be involved"

Not sure what you are asking - obviously the best/cheapest way to travel is by underground or train. Are you asking if you will need some form of car transport to complete the journey or that because you have large amounts of luggage you can't travel by underground ?