Some good advice here. Whatever you do make sure your wife is on board with you. Also make sure that your route is safe enough that she'll not worry about loosing you. Tri and similar pursuits eat up a lot of time and money so enter cautiously. That said, I think it's important to have sports or hobbies to enjoy and I've known many kids who've gotten in to bike racing or tri through their parents. Might not be a bad idea to wait just a bit though until they're a little older and don't require so much time from you and your wife. Also consider doing rides with your kids where you can get some exercise and talk to them. Highly recommend a bakfiets or seats up front (mounted on top tube just behind handlebars is my favorite) for this. I don't know many kids who enjoy staring at their parents sweaty rear for any length of time.

My commute varies from 2 miles to 11 miles each way with a few hills in the longer distances. I do it all on a city bike. Very comfortable, carries stuff, and no extra time spent changing clothes, showering, etc. If I had to go much farther than maybe 15 each way I'd likely consider a road bike.