We had Oyster cards from our visit in January and were easily able to top them up and hop onto the Piccadilly line at terminal 3 in Heathrow. We then change at Hammersmith station to the District line and get off at Aldgate East station, which was just a few short blocks from where we were staying. We were travelling on a Sunday when the tube cars were not crowded, so we had no problem with room for our bags. I had only a carry on and my husband had a bag only slightly larger than carry on and a small backpack. By keeping our luggage light, we managed to save a lot of money by not taking a taxi or car service and the tube is even cheaper and for our destination, faster than Heathrow Express.