People annoyed on the underground... well I don't think anyone can avoid that.
If you aren't totally familiar, each station has a wheelchair/stroller/luggage gate near the stiles, so you can get in and out with large pieces of luggage. The worst case is that you have to wait forever for a lift in some stations.
The cheapest is to take Picadilly line, but Heathrow Express is much easier to navigate with luggage. As long as you don't mind moving slowly, I wouldn't worry about the others.