I mean that a smart phone has the ability to access the Internet as one of the differences between it and a basic normal mobile phone.
That depending on your account with your phone service provider they will either give you a data allowance to use for accessing the Internet or not. But when your allowance is all used up for the month, it is the same for both; you pay more for further Internet access.
In the phone settings menus there will be something you can tick to turn off & on whether you use your phone service provider as the way to contact the Net. I'd advise turning it off until you are confident you know when your phone will access the Net. It allows you to control when it uses up your allowance or costs you more, as appropriate.
In addition there will be a setting to turn off & on the WiFi detection. If you are near a wireless modem (for your PC internet access) or near one you know the password for (some public places provide access for "free") then you should be able to experiment on how to select one of the WiFi networks your phone detects, and logging on to it. Once done, whilst you remain close to the modem, your phone can access the Internet via it, for no further cost to you, rather than paying your service provider for access.
But without someone there to show you it is a case of playing with the menus and trying things, I think. Hopefully the phone user guide will help you.