You cannot legally access the internet on a PSP for free without an existing internet connection. The PSP requires a legitimate Wi-Fi network (home, public hotspot, phone tethering) or, for the PSP Go, a paid 3G data plan (discontinued).
The PSP's browser (NetFront) is outdated and cannot render modern websites. For practical use, a smartphone is vastly superior. If preserving a PSP for homebrew, research reputable communities like r/PSP for safe CFW guides (e.g., Infinity, PRO-C2).