Laptop doesnt find internet access with route

Hello,
I have a laptop (Toshiba ...windows vista) it is WI-FI. I bought a router ( cisco linksys..wireless-G broadband router) so I can use the internet. My laptop is able to sign onto the internet obviously using my neighbors router etc. When I connect my router to my laptop it is able to find the router and signal is strong yet it fails at internet access. Please explain to me what I may doing wrong. Also, If internet assecc works I do not want my router to hooked up to my laptop meaning: the blue cord. I would like to be able to access internet in all rooms of my home without having to carry around the wires.