There are quite a few of us here that have sailed or are sailing with young children. You've come to the right place!
Your questions and hesitations are certainly valid and understandable. But the short answer and good news is: Your kids are a perfect age for getting out on the water.
As far as addressing sail and boat handling under way with kids aboard, you will find methods that work for you. In my opinion, an important part of making this successful is to begin with "baby steps". In other words, start small, acclimate the kids (and yourselves) to the boat before shoving off on an extended trip. Try some short day trips first, for a couple hours at a time.
Maybe begin by motoring around and familiarizing your family with the home waters and how the boat handles/behaves/moves. Next time out, motor upwind for a bit, hoist or unfurl a single sail, point the boat downwind, cut the engine, and ghost along for a while (i.e. don't make your first sail a long beat to weather). Stow sails, motor back. Maybe drop anchor and enjoy a picnic.
Spend your first night aboard at the marina slip/mooring. That way, you can get used to sleeping aboard with a lifeline to shore in case you've forgotten some necessity. Pack quick meals and easy snacks, so you can spend more time sheparding the kids and less time with "chores".
Make sure your kids have and wear properly fitted PFDs at all times when on deck/dock/dinghy. This firm rule will go a long way to easing your nerves, knowing that you will have time to react/intervene if one of the kids goes overboard. In another thread I recounted the sad story of a young family that lost their 2 year old overboard in a squall, who was not wearing a pfd. Learn form this and make the PFD rule ABSOLUTELY non-negotiable. Toddlers especially will fuss about wearing PFDs, but they eventually acclimate if you invest in a good quality/properly fitting life jacket(hint: Mustang L'il Legends).
I will also say that I think you've chosen a great starter boat for your young family. Catalina 30s are the Honda/Chevy of sailboats and it's a good fit for many coastal sailors that are starting out with young families. There are some known "issues" with older versions of that particular model, so be sure to check out the info we have here on Sailnet before completing the purchase (survey a must).
I'm a bit all over the place here. Sorry about that -- in a hurry. I will try to dredge up some old threads where folks have asked questions similar to yours. And I'm sure others will chime in as well.
P.S. Despite what some might suggest, kids and boats go together like milk and cookies. Trust me, I speak from experience. We had three kids under the age of 4 when we got our first boat (two in diapers). Boating is a big part of their life, and has enriched them in countless ways. If you can give your kids that experience, you are giving them a true gift that they'll carry with them all their lives.