Both cities are served by Metra trains. However, it's two different train lines.

It's a tough call. You can take a shuttle or cab from your hotel to the Arlington Heights Metra station - metrarail.com/Sched/cnw_nw/cnw_nw.shtml

From there, take a train to Barrington. But if you look at the map, you're traveling east to go west. You can also take a cab to the Palatine station. But you're halfway to Barrington at that point.

To be honest, it makes more sense from a time and practical point just to take a cab each way. http://www.americantaxi.com/