@SherylJones - I don't think you do need and landline or special TTY machines; I think there are cell phones you can use. CNET, for instance, will let you search for reviews of TTY-compatible cell phones.
I have only once received a call from a TTY relay service. It was actually really disorienting; neither the relay operator nor the person calling really explained what was happening, so it was hard for me to figure out what was going on (I did after a minute or two, naturally). This was at a job I had once where I was taking phone calls for job applicants.
Maybe TTY relay operators should have a standard spiel that they can use for call recipients who are not familiar with the technology. But this was several years ago, so maybe now they do? Or maybe I just had a particularly awkward experience.
I do wonder to what extent TTY will continue to be a technology that's commonly used by the deaf and hard of hearing now that so many people have a device in their pockets that does something similar - a cell phone capable of text messaging!