Who do I contact to find out what type of sewer drainage I have? Or is there another way of finding out?

I would start with the City.
If they don't have your address on file they can do a dye test. They put some dye in the sink and see if it shows up in the town sewer.
Do you know if there are municipal sewers on your street?
If you are out in the country you are probably on septic.

The city should have a listing of all sewer connections and even their locations. Also at least here the county has a listing of all septic systems and even their field layouts since you have to get them approved and inspected before you use them.