Panama City is a great place to live.

I moved here about 10 years ago. I have a 2 1/2 year old and a 3 month old, and I used to work at the college. Students who start at GCCC and then transfer to four year institutions do better in their junior/senior years than students that start at the four year school... so GCCC is a good school.
You can find school grades on the Bay District Schools website... And daycare info is found at the Early Education and Care website. Of course, you'll want to check them out in person, too, but it's a place to start.
There are some playgroups for your 2 year old... find them on meetup.org and Yahoo Groups. Do a search.
Downtown PC is undergoing a resurregence... Lots of small festivals on Harrison Ave on the weekends. Every month (first Friday) there is Friday Fest where they close off the street and have music and food and old cars.
The beach is hopping most of the year now... especially with Pier Park opening up. I live on the beach (you always say "on", not "at" here) and know the schools are good here. Patronis and Beach Elementaries are well established and Breakfast Point is just finishing it's first year. The new Publix just opened up yesterday on Back Beach Road at Breakfast Point. There are several new apartment complexes too -- two on Alf Coleman Road and one on Back Beach by the Catholic church. The area around South Thomas Drive (parallel to the beach) is predominantly rentals... It wouldn't be my first choice for a place to live with kids. Without knowing your budget, I can't steer you in any particular direction.
Pick up a Coastal Kids Guide (they're everywhere) to find out about what to do with your kids. After school care is usually provided by Bay Base (at the local schools), Boys and Girls Clubs, or Girls Inc. I use Dr. Ingrid Rachesky as a pediatrician -- she's extremely thorough and competent. This is the third ped. I've been to... and I'm sticking with her!
You can buy consignment kids clothes at Doodlebugs on the corner of Airport and Lisenby Road. Another kids consignment clothing store is opening in the Publix Plaza on 23rd St.
Most of the area libraries have summer programs for elementary aged kids and year round programs for the under 5 set. There is a brand new library on 11th street.
In town, get your doughnuts at Dan-D-Donuts across from Bay High School on Harrison Ave. On the beach, it's Thomas' Donuts in Laguna Beach on Front Beach Road.
Grocery Outlet and Save-A-Lot are cheap, but scary. Publix is the cleanest and stocks the most variety. Winn Dixie can be more expensive than Publix on some things, but has the best meat. WalMart is building a new supercenter in Lynn Haven, and will close the one on 23rd St. when that opens. The other two WalMarts (Callaway and PCB) are supercenters, too, so they have groceries -- best prices!
Los Antojitos (on Hwy 98 near the college) has the best Mexican food! Also, show your student id at most of the restaurants around the college and you'll get a 10% discount. Sports are big at the college -- the Lady 'Dores basketball team has won the national championship several times and almost always goes to the tournament. The softball team is also very good. As a student you get free admission.
Hope that helps some!