As long as it's a caching proxy server (proxy doesn't automatically mean caching) then most websites should be sped up, because things like graphics won't need to be redownloaded or checked over your connection (even if your browser doesn't actually download them again, it still tries to contact the server to make sure they don't need to be). Games won't be affected, because for one thing, the proxy should only be set up for http traffic and your web browser configured only to use the proxy for port 80 web traffic. Nothing else will even contact the proxy. Secure pages should be https, so your browser will only use the proxy for those if you specify it should in the settings.
Make sure your proxy is configured to ignore traffic coming from outside of your local LAN, otherwise it'll have a chance to be hijacked and used by people outside the network (unless you're behing a NAT router).