If you're looking for a meeting place or a good stop around ½ or halfway between Tarrytown and Springfield, you can try New Haven, CT, which is almost an hour east of Tarrytown, or about 1 hour south of Springfield.
Another possible place to stop near the midpoint is West Haven.
About ¼ or a fourth of the way, you can stop at Norwalk, CT. This is about half an hour east of Tarrytown.
About ⅓ or a third of the way, you can stop at Bridgeport, CT. This is almost an hour east of Tarrytown.
About ⅔ or two thirds of the way, you can stop at New Britain, CT. This is about 1½ hours northeast of Tarrytown, or about half an hour south of Springfield.
About ¾ or three fourths of the way, you can stop at Hartford, CT. This is about 1½ hours northeast of Tarrytown, or about half an hour south of Springfield.
If you drive straight through without stopping at all, the total distance is 122 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 2 hours, 1 minute.