Our city provides free wireless access. If I'm near the window my USB receiver is fine, but if I go to the center of the room it dies. Basically I want to put an antenna on top of the house. I'm a total noob. Can somebody tell me exactly what hardware I need to bring it down with a cable (and what kind) about 30-50'?
If the connection is fine near the window, I wouldn't bother with anything nearly as complex as run cabling to an antenna on the top of the house. That's just unnecessary. Instead, use a wireless repeater. The repeater is placed near the window, connects to the remote wifi as a wireless client, and repeats the wireless signal from the window to your PC/laptop.
Since you’re using the city wifi, you also have to consider security. Just as when using any open wifi system, your communications are “in the clear” unless using SSL or a VPN. That’s why using the city wifi is a mixed blessing; cheap but comes w/ some risk. You might want to consider using a VPN service to protect your privacy.
Also, assuming you’re only connecting the one device (PC/laptop), make sure you have your Windows firewall up, and disable all file/printer sharing. Remember, you’re sharing your PC/laptop on the same network as other strangers! If you intend to support multiple clients, then I would find a repeater which supports a “routed bridge” to the remote wifi (aka, WISP router). That will provide you with your own firewall, your own subnet, your own DHCP server, etc. A dd-wrt compatible router is well suited for these purposes.