If you're looking for a meeting place or a good stop around ½ or halfway between Waterbury and Bristol, CT, you can try Wolcott, CT, which is about 11 minutes northeast of Waterbury, or about 11 minutes south of Bristol, CT.
About ⅓ or a third of the way, you can stop at Waterville, CT. This is about 8 minutes north of Waterbury.
About ⅔ or two thirds of the way, you can stop at Hancock, CT. This is about 15 minutes north of Waterbury, or about 8 minutes southwest of Bristol, CT.
If you drive straight through without stopping at all, the total distance is 12 miles. The distances shown above don't include local roads to get into each town, so if you're planning a long road trip, make sure you factor in traffic and road conditions, rest stop breaks, and interesting detours along the way. If you're in a rush, you can do the trip with zero stops, and your total travel time would be around 23 minutes.