We can send out SMS to individual contacts (or from other custom objects ). To do so, all you need to do is edit the page layout of a record (i.e in SF classic view) and add the "Send SMS" button under Salesforce1 Actions section and save the layout. Once this is done, you will be able to send out single SMS in Lightening Experience.  


Please refer the below screenshot for reference:

The Send SMS button appears on the page layout of the record:

Once you click on the Send SMS button, a pop-up appears where you can select/enter the appropriate information, which enables you to send out a single SMS. 

Once you click on the Send SMS button in the above pop-up, it will show you the notification of the SMS that was sent out.

Similarly, you can setup Bulk SMS button on the object and use it accordingly.