The bus station in Neily is near the end of town - just follow the road you came in on until you see the station on the left, or take any major street straight back (going away from the highway). The number for Tracopa, the bus company that handles most trips around this region, is 2 783-3227.
It's pretty easy to hail a cab here, and there is also a private transport company, DILAN, located ocross from the bus station, Tel: 2 783-2121.
There is a SANSA office in town if you want to arrange flights in other cities or fly out of the Coto 47 air strip, 4km south of the city. 
BUS SCHEDULE
Neily to Paso Canoas - $0.75 (350 c); 30-40 min.

Half hourly, 6a.m.-6p.m.

Neily to Golfito (Buses Golfitenas) - 1-1.5 hours
a.m. 5:45, 6:15, 6:45, 8:15, 9, 10, 11
p.m. 12:30, 1:30, 3, 4, 5
Buses also stop near the bar El Rancho at 6:45 a.m. and 7:15 p.m.

Neily to San Jose - 8 hours
a.m. 4, 5, 8:30, 11:30
p.m. 5