Mine started walking at 13 months and straight away wanted to be walking outside, going somewhere. so as an activity we used to walk to a small local supermarket with the aim of buying just one or two items. It really wasn't about needing to buy shopping, it was about introducing him to a quiet shop (not our big one where we do the real shop), letting him experience it and also starting to teach him what behaviour was expected. It was great, he loved it and even though he is still quite young he has learned how to walk around for much of the big weekly shop in the big supermarket, helping get the fruit and veg etc into the trolley. It's so much easier to put the time and attention in when you don't need to buy anything other than perhaps a carton of blueberries to eat on a bench outside before walking home.
We also did 'trips' to road works
He loved to just stand and watch the diggers. Trips to train station, bus station, an airport, car show room, white goods store. It amazing what kids are interested in.
I hear you on the times of the groups. I always avoid groups that run at nap time, I just don't think its worth upsetting the EASY.