SMS and MMS are services provided by the carriers, and based on their (cell) network technology.
If your plan doesn't include them, you won't be able to use them.
Traditionally, they were offered as pay-to-play features, with either a per message charge, or an allotment basis.
As instant messages became a popular form of communication, bills started going up, which gave rise to alternative messaging apps, like AIM and Yahoo Messenger in the old days, to the multitudes we have today, like Facebook Messenger, Google Hangouts, iMessage (on Apple devices), Skype, Line, Telegram, etc.
WhatsApp became incredibly popular in certain parts of the world because it was a good app that enabled users to send messages without incurring any charges, and bypass the paid SMS/MMS services. That was worth $19B to Facebook, which bought the company to gain access to its users (and their data).
Such apps aren't based on the cell provider's technology, but simple IP, which can be carried over any network connection.
Most users now have one, or more of these apps on their devices and computers, so your best is see which ones are popular with your customers and install them on your new phone to use for messaging, as well as picture sharing; many of them even offer voice calling. It is unlikely now to find someone who strictly relies on SMS/MMS, so it's just a matter of finding the matching conduits among the other options.