Let me start by saying I formed my opinion before reading everyone's replies and came up with Brighton. For the vibe, the schools etc. We are a white family that lived in the city of Rochester up until 1 years again when we relocated to MI.
Are you familiar with Great Schools
GreatSchools - Public and Private School Ratings, Reviews and Parent Community
It gives a grade for schools which I take with a plus or minus but it also gives the racial make up of a district and each individual school as well as number of students receiving free/reduced lunch.
Back in the day Brighton was THE ritzy suburb but really that goes more to Pittsford/Fairport/Penfield and Victor (IMHO). Brighton does have some folks with money but I see it as being more academia than the others where there are more Doctors/Lawyers/corporate executives etc. This is a gross generalization and observation. As for diversity, again look at the greatschools breakdown.
Brighton is a relatively small district (which appeals to me), the town itself has an older feel, and this is referencing the 12 corners area (where Winton Road/Elmwood Ave and Monroe Ave converge. I personally would recommend this area of Brighton for its walkability and proximity to Public Transporation. And while I know some people in the Rochester area that don't drive they are few and far between especially with kids.
Going back to ethnicity I know plenty of white families and non-white families that are happy in the city and plenty of white and non-white families that aren't. And it usually has to do with the schools and one's own experiences.
As for Brighton we have had many "city" friends move to Brighton because they didn't want to lose the city feel and I don't know any that weren't happy with the move.
I hope this helps and I agree with renting but I also know with kids you want to try and get it right the first time so you don't have to change schools again. We were lucky when we moved here. We rented because we wanted to have time to figure out where we wanted to be and we ended up buying a house 1/2 mile from where we rented. But it gave us time to figure out what we really wanted.
I was very unhappy with Rochester City schools but that being said if you navigate right I know some people that are very happy, but it takes a lot of energy. there are some very nice and safe areas of the city where people go out at night and kids can play outside (there are plenty that aren't but more that are than people think).
I would also suggest getting on Nothnagle.com to look at houses to get an idea of price points and taxes. I think it is much more helpful than Zillow.
Good Luck
PS Please don't be put off by those folks that try to ruin any Rochester thread and turn people off. Just skip over those. That's what I do. I'm not saying people can't have an opposite opinion but there are some that go out of their way to not give any constructive criticism and just slam.