I like the city (a lot) but the first ring suburb doesn't sound much worse, for a lot less money.
Two thoughts:
1. Amenities are nice, but you want to think about the ones you will actually use in a few years. For my part, I want to be able to walk to 1-2 decent bars/restaurants (mostly so I have the option to drink w/o driving, I rarely go), a good park and a grocery store. Grocery store is probably the most important one on there, followed by park, but think about your list.
2. School ratings do a much better job telling you how rich the neighborhood is than how good the school is. If you are mainly thinking about moving for the schools, try to talk to parents, meet with teachers, get a tour, etc. before making a decision. There are obviously exceptions (you don't want your kids going to a school filled with violence or enough disruptive kids that the teachers can never teach) but in general your kid's success has a lot more to do with you than it does with the specific school
All in, if it has the amenities you need within walking distance I'd probably go (very close in) suburb in your case.