UX Booth recently did a posting on tools that help with web accessibility. I ran my personal website through all of the tools and found issues. (Special joy.) I particularly found the image analyzer from Juicy Studio to be quite informative. They have some other tools and links available on their website.
Foundstone (a division of McAfee) hosts a bunch of free tools on their website. These include tools that you can use to learn and practice security testing (like Hacme Bank). If you're new to security testing, consider spending some time with these tools and learning utilities.
From the Windows command line, you can use netstat to displays information on the status of the network and established connections with remote machines.
From the Windows command line, you can use route print to display the routing table. The command route serves to define static routes, to erase routes or simply to see the state of the routes.
From the Windows command line, you can use tracert host to show the route that packets follow to reach the machine “host”. The command tracert is the abbreviation of trace route, which allows you to learn the route that a packet follows from the origin, (your machine) to the destination machine. It can also tell you the time it takes to make each jump. At the most, 30 jumps will be listed. It is sometimes interesting to observe the names of the machines through which the packets travel.