You have a baby. You need to move somewhere with good schools. That should be your starting point. The good schools / nice area / cost equation is going to be hard enough to balance, without any added complications.
I'd be methodical about it. Identify the type of place you're interested in (size of town, proximity to coast, countryside, major cities, people you want to visit, transport links) and your constraints e.g. jobs, house prices. Get a map out and draw lines on it, identify the possible zones, then the possible towns / areas within those, then do some research on house prices and schools for each area. That should give you a shortlist to discuss, then visit and dig deeper into the top few.
I'd find the idea of randomly, or otherwise, alighting on one place and discussing only that, stressful and frustrating. It's so, random! How can you make major life decisions like that? I appreciate many of us move for jobs, partners, family etc so do end up settling in places we mightn't have chosen, given a clean slate. But you are in the fairly rare position of having a clean slate. Take advantage of that freedom!
The thing about wanting to escape some user 'friends' bothers me. Maybe you've stumbled across an exceptionally bad bunch but generally, relationships are two-way and for this to have happened, without you being able to assert yourself and stamp it out, you'd have to have participated in the development of those relationships. You can't move away from your own personality. Do you feel you've matured and outgrown these people, that you wouldn't make friends with similar people again?
It's much easier to drop old friends and make new ones in a city, than in a small town. If you are a bit of a pushover, a people-pleaser, that's going to work out worse for you in a smaller community.