Catskill is about 115 miles north of New York City. By car, take the New York Thruway (I-87) to exit 21 (Catskill) and cross the Rip Van Winkle Bridge, or take the Taconic State Parkway north and cross at the bridge. The drive takes about two hours without traffic.
By train, take an Amtrak service to Hudson station (on the opposite bank of the Hudson). From Hudson, a short taxi or rideshare across the bridge brings you into Catskill in about ten minutes. There is no bus service from Hudson station, so arriving without a car requires planning ahead. Several New York City coach services also run to the Hudson Valley on weekends, though stops are generally at larger towns.
Having a car is strongly recommended if you plan to explore the Catskill Mountains or visit Olana and other surrounding sites. Catskill village itself is walkable, but the rest of the area is not.