I have read through the FAQs and searched around the internet, but can't find the answer to my question. Hopefully I won't sound too stupid here.
If I use a router and someone finds my IP address, that will initially at least, be only my routers IP address, not the internal IP address of my computer. However, using that would they be able to find my internal IP address? How would they be able to find it? How could they make the jump from my router to my PC?
Just knowing the address of a router on your network/subnet, does not mean that someone can get your IP address. But they can narrow it down substantially. eg if the router's address is a class B address and you're using subnetting (most class B networks do) it's a safe bet a person could narrow down your IP to within about 254 hosts (assuming you're using the 3rd octect of the address for subnetting).

But this isn't how TCP/IP works. IP packets already contain the source address of YOUR machine. Router addresses (not IP) are only used at the Data Link level. So if you're already connected to someone else box (assuming it's TCP/IP connection) they already have the IP address of YOUR machine regardless of the address of your router/firewall (assuming you're not using a gateway/proxy).

In English: A router routes packets. It does not proxy/mask your IP address.

And if it is done with "scanning" my router somehow, exactly how do they do that? Is it a hacker program or just commands? I've tried it from my friend's computer to my own and I can't figure out how that would work(of course that means little).
A person can get your IP address using simple network commands. eg netstat

I'm trying to make my computer secure and an extremely knowledgeable friend of mine told me that routers are better than firewalls. But I don't want to screw myself here. Should I install a firewall as well?
This is a gray area because some routers contain firewall type functionality but I believe that routers should route packets and guard against DOS attacks at the data link/network layers. Firewalls should look after the rest.