I will echo the learning to sail with kids on board. Not a good mix. I went through it with young kids when I moved up to a keel boat. The stress of getting in and out of the harbour, right of way issues, squalls, these were easier to deal with without the kids to worry about.
The suggestion of buying a trailer sailor is a good one. 22-26ft. learn, bump into things etc.. take the kids for brief daysails in fair conditions once your confidence level is up.
As stated before, you will also fairly quickly develop a list of must haves, don't need etc.. for your upgrade. This will help tremendously when you iwill be choosing the right boat. (you will also lose very little on the trailer sailor).
Finally when that happens, your triplets will be walking and will need car seats on board as well....