I am creating a network with 6 subnets. I know I need to connect the subnets to a router in order to for each subnet to communicate with each other. My question is, can all the subnets be connected to the same router? Can this router also be used as the gateway to the Internet?
To connect up the subnets to a single router, you will need a business-class router that can handle multiple LAN zones. Most basic routers are only for a single network range, but you can find many firewalls and more expensive routers that can manage multiple networks. For instance, on the Sonicwall TZ 100 series firewall and greater, you can configure each physical ethernet port to an independent network, which can each be the default gateway for a subnet.
If you're wanting to have six subnets, you'd need to look into a Sonicwall TZ 215 or NSA 220 firewall to have the number of ports needed for that many subnets.
The only way not to use VLANs in this situation is if each subnet has their own separate physical switch, which can get kind of confusing and costly as well. But yes you can do this without VLANs or with VLANs, it comes down to the network switch infrastructure that you are using.