The bus station and train station are opposite each other. They are within walking distance from the old town area. You can find them on the junction between Ulica kralja Tvrtka and Ulica Ante Starčevića. You can find it on Google Maps here..
If you have GPS on your map and a spare 15-20 minutes, it is an easy walk to find the coach/train station.
A number 2 or 4 bus to/from the center of Zadar will take just five minutes and costs 10 kuna.
Taxis to/from the old town/center of Zadar to the coach/train station should cost 20-40 kuna. Be sure you agree on a price before you get in a taxi.
There are many domestic busses every day, especially to other big cities such as Zagreb and Split.
The train station (which can be found opposite the coach station - see above) has a few (slow) routes to some other cities in Croatia.
Trains are not as popular as coaches, as they are slow, sometimes have no aircon and are not comfortable. But you will have the advantage of looking out of the window and admiring the views.
A train to Zagreb will take about 5.5 hours (or 8.5 hours on overnight trains). A train to Split will take about 3.5hours.
Check online train timetables at Hzpp.hr.