Yay! Welcome to Cincinnati! We moved here about 2 years ago and we love it!
There is so much fun stuff here to do with kids. I have a 4.5 year old and a 2 year old, both boys. We love (LOVE.) the Hamilton County Park system
. The Museum Center is fantastic - three very interactive museums (Children's, Natural History/Science, and Cincinnati History) in one. The Zoo is a pretty great one. Depending on where you live, there are plenty of great city parks around, too (in Cinci proper there's Eden Park and Ault Park, and you'll probably find plenty smaller in your neighborhood - we're in Montgomery, and there are 4 within walking distance). Right now is good farmer's market season, along with some decent apple picking (we go to Iron's Fruit Farm in Lebanon). Pumpkin season around here is fantastic.
For event info (story times, kid-centered and kid-friendly events), pick up a Cincinnati Family magazine somewhere around town. Between that and the County Park event booklet, I plan what we're going to do throughout the month.
I really can't recommend the Park system enough. We spend half of our time at one of them.
Welcome and enjoy!