I'll take a bit of a contrarian view here. What I would do is go and buy a boat in the 30-32' range that is stable and accomodating for the family and will be MORE stable than the smaller boats AND easier to handle...(I know that sounds strange but the bigger the boat the EASIER it is to handle since the added weight makes it handle bigger seas and heavier winds without stress). The only thing that is MORE difficult about handling a bigger boat is coming into the dock UNDER POWER and that just takes practice.
Next I would HIRE a captain to go out with you a few times until you feel comfortable with the boat. This is better than learning on a small boat and then having to re-learn on a larger boat.
You can easily get a NICE Catalina 30 for about $20K which is way under your budget and another grand for a captain to teach you how to sail YOUR boat and work all the systems is a great investment. When you're through, you'll have a nice boat for local cruising with the family with a real head and galley and even air conditioning and a dockside electric system so you can spend weekends aboard in comfort.
Of course...fit out the boat for single-handing and get an auto-pilot etc. ...but don't be afraid of starting on a larger boat.