I've been off and on the car ownership/usership during the years - so I also have quite some experience carrying heavy things by public transport.

Once upon a time we were travelling from Russia to Norway with one 6yr old child, one newborn baby, lots and lots of luggage ... my dad-in-law simply couldn't drive us, because we had too much luggage to fit into the car (besides, we left in the daytime, so he was at work anyway)! So we managed all the way from the suburbs of St.Petersburg to the suburbs of Oslo using only public transport. We had all the luggage on wheels, no problem on flat ground ... but the stairs were a big problem! First metro station in St.Petersburg, one of the only metro station being above the surface ... we had to climb stairs. Then we had to change metro ... and climb more stairs to get from one metro line to the other. We took a bus and came to a ferry terminal ... and to enter the ferries, again ... stairs. We arrived in Stockholm and was actually staying for a full day in Stockholm without any stairs at all. Train to Lillestrøm and Grorud, and there again ... stairs!