Wireless Printer Help

What does "another network" mean? You installed an access point, a router, or what?
How is the printer connected?
Can you disable the wireless portion of the ISP-supplied modem since you aren't using it?
New router? It looks like this misconfigured device should have been set up as an access point, not a router.
You forgot to mention how the printer is connected and while you're at it, you forgot to mention why you care about how print jobs make it from your computer to your printer.

I'm jealous.
Presumably, the other network is just an ethernet switch, not another router. What box do you have installed at this "other" network? Maker and model please. If this box is another router, you goofed and will need to replace it with an ethernet switch.
Actually, it's probably just going from one of the ethernet ports on the back of the unspecified make and model router, to another port on the same router. While strictly speaking, it's going "through" the router, it's really going from one part of your LAN (local area network) to another part of the same LAN, and not through the router from LAN to WAN (internet).
It's done all the time, but you don't need it here.
Yes. If all your devices are on the LAN side of the router, with all the IP addresses in the same class C IP block such as 192.168.1.xxx where xxx is 1 thru 254, then everything on the LAN side should talk to everything else on the LAN side.
Same as #1.
Hint: If you post a question with such a vague description, you'll get equally vague answers. Make and model of all the boxes involved, including the printer, might be useful to supply a more specific answer.

Dear CR,
I am new to networking, but I was still able to connect two routers wirelessly using the information on the DD-WRT Wiki site. In case you don't know, DD-WRT is an alternate firmware for routers, and if you never heard of it, you probably don't have it on either router. Nevertheless, their Wiki is a great source of information about all sorts of networking issues. It sounds to me that you need to start at the page about linking routers:
It will direct you to other pages with more detailed information about various ways to do the linking. Even if you don't install DD-WRT on either router, you can probably adapt the information to the OEM firmware for your routers. Once you get control of linking your routers, it should be easier to get your printer to work the way you want.
Neil