If your psp go console doesn't connect to the internet successfully through a wireless router with a active internet connection, first you will have to identify what is the cause to the problems.
Below i have underlined the cause and below it i have listed the solution on how to fix it:
My psp go can't find my wireless router?
If your psp can't find your wireless router, make sure the Wi-Fi router and the Wi-Fi on the psp go is switched on. Also make sure you Wi-Fi router is in infrastructure mode, if the wireless router is in adhoc mode, your psp go will not find its access point name, as the psp go can only pick up Wi-Fi routers that produce a infrastructure signal.
My psp go has found my Wi-Fi access point, but it will not allow my psp go to connect to it?
If your psp go will not allow itself to connect to the wireless router, the first thing to make sure is if your Wi-Fi router is connect to a live active internet connection. The second thing why the psp go could be failing to connect to the wireless router internet access point it could be because it is password protected. If it is password protected, then insert the password into the access point on your psp go console and the problem should be resolved. Also make sure the psp go has a minimum of 35% of signal strength from your Wi-Fi router, as below that percentage the psp will fail to connect to the access point and as a result your psp will have a connection error.
The internet is not supported on my psp go?
If the internet is not supported on your psp go, then i suggest you to change the type of security encryption you are using on your wireless router to WPA or WEP, but only if you using types that are not supported by the psp go, such as using WAP.