M.W.
Bellevue Square has a nice play area, up on the 3rd floor away from the stores. There's a Gymboree/KinderMusic up there now too. Very large area. You can also do Gymboree classes in Ballard on Market Street (toward the Locks). Most of the Seattle parks department community centers have classes for little ones, but we discovered that they are usually for 2 or 3 and up. The Ballard community center does have a "drop in" play center that's $2 per visit or something like that.
For walking shoes, we found that the See Kai Run shoes worked best--mostly because our daughter has tiny feet and needed size 3s! They were the only outdoor/soled shoes that would fit her. Me N Moms in Ballard sometimes has discounted past season See Kai Run. I also like the Robeez Tredz shoes--but they ran big so she had to be a little bigger before we used those.