You guys are all talking out of your ass, or at least being ambiguous, just so you know.
Generally speaking, switches send packets along with regard to their layer 2 headers, and routers concern themselves with layer 3 headers. If you don't know what the hell I'm talking about when I say "layer 2" or "layer 3", you are not qualified to say what a switch does or does not do. Some switches are called "Layer 3 Switches"; they're routers. Throw in a firewall, NAT/NAPT, and a slick management interface, and you've got your linksys router at home.
Yes, something like a Cisco 3750 can do DHCP, but from an infrastructure standpoint you really want your switches switching and your routers routing, not handling IP Address assignments or doing other service-level crap.
You do not need a modem to get on the internet. All you need is a connection to a router that's showing you non RFC1918 addresses (and an admin that's not stupid enough to assign non 1918 address privately), and you're on the internet. Generally, modems do this job using a specialized line protocol like DOCSYS, but there's no reason you can't do Ethernet from endpoint to endpoint.
For what it's worth, my Linksys WRT54G (Custom Firmware) is a managed switch, with 802.1Q support on every port, it can do STP, and QoS with packet inspection. (
http://www.dd-wrt.com/). This is actually the reason Linksys (aka Cisco) has stopped making their routers so easy to flash with custom firmwares, as the functionality provided by them rivals some much pricier Cisco devices, with the only real downside being a loss of performance statistically equal to zero in smaller networks (/24 or smaller).
Sorry to bump the topic off, though...and I'm also not trying to knock anyone down, here, I just love talking about networking
. We do need to see the network topology going on here, as it sounds like the network is the problem.
What connects you to the internet?
What connects your computer to that?
How are your computers connected to that?
Can you provide model numbers, where possible?