How do i install a network where database can accessed between cities?

Hi!
Probably you'd like to know before hand the budget you have available for this and the availability required.
For example, if its a high budget high availability project, I would just use dedicated links between the locations (if those are customer premises)
If you have low budget, consider using vpn tunnels (might be a little slow)
if you are setting up something on windows... probably samba/ms shares accessable thru the internet/vpn will help.
if you are gonna use a SQL database or somethign like that, probably the best chance is to use a linux server with a dedicated link
good luck with your network!

is exposing the database server directly on the internet.

VPN or dedicated links.
Another thing to think about is redundancy, depends on how critical the app is, and your infra-structure. Put it in one place and you have a lot of single points of failure.

You might be better off hosting it at a datacentre, or having local servers and some sort of replication scheme.

Not really a job for a novice this, you need to get definitive answers on criticality and availablity before you even look at tech.