The Raspberry Pi is the nice device to set up a small server at home. It is cheap and there is a great community for it to get help. There are tons of things to do with it and the first step is always to get the system up and running.
Raspbian
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If you care about privacy and freedom on the internet you should consider using and supporting Tor. If you have internet bandwidth to share you could easily support the Tor network by running a Tor Relay.
The so-called Tor (or onion) network is a way to get better anonymity in the web by hiding your IP address. A really nice option is to use the Raspberry Pi to open a new WLAN network and act as a gateway into the Tor network. All traffic from the devices connected to this WLAN will be routed through the Tor network.