Setting up 2 routers and a switch

Did you set the routers default gateway correctly?

Only one IP address is required, but the network needs to be properly configured. If the Internet IP address is dynamic (very likely based on the initial post), then the modem can't be connected directly to a switch. It has to be connected to a router with NAT capabilities.

Modem <---> Router 1 <---> Switch <---> Router 2 (can also be connected directly to Router 1)

This way Router 1 will provide access to the Internet. The Internet connection on Router 2 requires a static IP address and it's default gateway has to be Router 1. As long as routing is properly configured, it will work just fine. I've had a similar setup (with added complexity) for several years.

Example:

Router 1
LAN IP address 192.168.1.1
NAT range 192.168.1.100 - 192.168.1.120
Internet IP address: Dynamic (provided by the ISP)
Route traffic to network 192.168.2.0 mask 255.255.255.0 to 192.168.1.2 (the other router) or use dynamic routing (RIP has to be enabled on both routers)

Router 2
LAN IP addres 192.168.2.1
NAT range 192.168.2.100 - 192.168.2.120
Internet IP address: IP address 192.168.1.2 (connected to Router 1 or the switch)
Gateway IP address: 192.168.1.1 (Router 1 IP address)

Even though you have a static IP address, you still need to connect a router to the modem, not a switch. The problem is that a switch doesn't do NAT and you absolutely need one to translate the single Internet IP address to multiple internal IP addresses.

What you're trying to achieve is very similar to what I have, but my setup is a bit more complex due to a corporate VPN. In your setup, you need to specify the IP address of the DNS on both routers if they are DHCP servers.