No, you're not doing anything wrong. You should be able to have cellular data turned off and be able to browse the web just fine if connected successfully to a WiFi network. Try disconnecting and re-joining the network. If you still can't browse, try to investigate the WiFi issue further - for starters:
-- Do other apps connect OK? Mail, others?
-- Test on another open WiFi network, at a coffee shop or wherever is convenient. This will help determine if it is an issue just with your home network or with all WiFi networks.
-- Try a reset of network settings.