I've currently got 15 computers plugged into a switch along with a cable modem providing internet access. They all receive their DHCP addresses from the cable modem. I'd like to segregate the computers so they can't get to each other. I thought I'd be able to do this with VLANs, but I'm new to managed switches and I'm having trouble figuring out the configuration. I've currently got a Dell Powerconnect 3524. Any help would be appreciated.
What you want is called "port isolation". Dell PowerConnect switches do this via Private VLAN Edge (PVE). Under the port configuration (near the bottom) is the selector for the "uplink". Unfortunately, it has to be a gig port, but that shouldn't be much of an issue. Attach the cablemodem to "g1" and set all ports to PVE uplink of "g1".
This use case is covered by Private Vlan. I only have direct experience with Cisco, however the documentation for your switch, Dell™ PowerConnect™ 35xx Systems User’s Guide, shows that this feature is available. Page 297 has the port options, the option that should suit your needs is on page 300:
Private VLAN Edge (PVE)— Indicates the Private VLAN Edge (PVE) group to which the LAG is configured. A port defined as PVE is protected by an uplink, so that it is isolated from other ports within the same VLAN. The uplink must be a GE port.
As I said, I don't have direct experience with Dell, so you should lab this.