You can connect your PS Vita device to the internet using your Home Wi-Fi access point or any public Wi-Fi Hotspot. Set up a wireless connection first.
Tap the Settings' icon. Now tap Start -> Network -> Wi-Fi Settings.
Choose the Wi-Fi access point you want to use. You'll need to know the SSID (name) and the password.
When the PS Vita successfully connects online to that access point it will remember it and automatically connect to it next time.