Just because your phone does not have a data plan doesn't mean it can't connect to the Internet. The phone's Wi-Fi service is separate from the data plan, meaning that as long as the Wi-Fi transmitter isn't damaged or turned off, you can still connect over wireless networks. Using Wi-Fi both at home and in public, there are several ways to make your data-free smartphone still useful -- and even to make calls.
Public Wi-Fi Networks
If you're using a public Wi-Fi connection, take steps to protect your data like using Virtual Private Network apps to secure your connection. VPNs encrypt data sent from your phone to an app or website you're connecting to, making it harder for hackers to steal private information. There are several VPN apps available for both Android and iPhone.
If your phone still has a voice plan attached to it, then regular phone calls should be unaffected by your lack of a data plan. However, most cellular carriers don't offer voice-only plans for smartphones -- Virgin Mobile is the only major carrier offering voice-only smartphones -- making it a little trickier to retain voice calling without data.
Voice Chat Apps
Apps that send voice calls over Wi-Fi connection can also be used without a data plan. For example, Skype and Google Voice both make it possible to call other users for free, so long as they are on Skype or Google Voice respectively. With a small monthly fee, Skype can even make calls to phone numbers, making it possible to completely cut out your carrier and adopt a considerably smaller monthly fee to maintain voice services.
The only catch is that you have to be connected to a Wi-Fi network in order to place or receive calls. While freely available public networks are more widely available than in the past, this does mean there may be times when you don't have coverage. Mobile Wi-Fi hotspots are one way to get around this issue, although they do require a purchase and monthly subscription.
While many apps on smartphones rely on an Internet connection, any apps that don't can be used anywhere without a data plan. When you do need your Internet-dependent apps, a Wi-Fi connection works just as well.