You connect to the router, but the router doesn't let you talk to the internet? No love with wireless, switch to wired and get internet, then switch back to wireless and you're still good.
I think you need to start with your router and with DHCP.
Get a wireless connection and go to 192.168.1.1 and see if you can talk directly to your router. R-click your network icon and check your connection information to see if you've been assigned an IP. If you haven't, you're not getting DHCP. If you're not assigned an IP, your router isn't letting you talk even though you have a "connection". Disable DHCP on your end by assigning yourself a static IP. See if that helps.
My theory is that your router is slow on DHCP to assign you an IP, and thus also isn't forwarding your TCP traffic upstream. Having a "connection" just means there is a link, wired or wireless, and the "strength meter" says nothing about the traffic on that link.