Other travellers are generally pretty good about lending a hand to parents of small children. The entry to the double-decker trains to Avignon is at platform level, then you have one or two steps down to the lower level or several steps up to the upper level. There is some luggage storage as you get on the train and more upstairs.

You should expect it to be very busy on Saturday, as it is the beginning of the Paris school holidays.

The Gare de Lyon is "separated" into two areas: blue and yellow. The track for your train will be posted about 20 min before departure, but the zone will show up before then. You should move to the correct zone as soon as you get there, as they are doing work and it can be a bit of a walk. The map in Tennisbum's first link shows the layout of the station and the zones.