No, you would not connect to your wireless router. That would make a circle – looping you back to your network, without any Internet access, and causing other networking problems. Newer Wave WiFi products, the firmware, has a feature allowing you to “Ignore” such access points and routers – to take them off the scan page and prevent you from attempting to connect to them.
You are generally looking for Public WiFi. The only exceptions will be if you know what you are connecting to – such as someone else’s boat that has allowed you or a Personal Hotspot that you own. You would not connect to your boat name and should not connect to things that are not public. Connecting to someone’s HP Printer would not get you Internet access.