Laptop w/wireless will only access Internet on home network

I’m not the most tech savvy person in the world, so bear with me. I have an HP notebook with built-in wireless. I also have a home wireless network set up. When I’m at home, my laptop always instantly picks up a signal and I access the Web with no problems. However, when I am outside of my home, away from my home network, whether it is in Starbucks or in a public library w/a free wireless setup, I cannot get online. It always tells me I’m connected and that I have a strong signal. But when I try to access the Web it doesn’t work.

Help?