It sounds as if the person who gave it to you/sold it to you didn't ensure that all of the drivers were properly installed. I'm guessing you're running Windows XP on the machine. The Device Manager has a list of all of the devices that your computer has installed on it, including the items that may be giving you problems.
If there are exclamation points next to any of those devices, it means that there's a problem with the driver or the driver isn't installed. You can click on the device and click Properties to determine what the problem is. If the driver just isn't installed, then you can use a computer (that's working) or plug in an Ethernet cord to obtain it.
Additionally, you need to make sure that any other software isn't interfering with Windows connecting to the internet. If the owner before you used a Cricket adapter to connect, it probably had its own software to connect to the internet. You need to make sure that's uninstalled/disabled, but it should have been if he restored the computer before you got it.