Cheapest way to get a phone is probably to ask around your family/friends/colleagues for somebody who has a spare handset that they no longer use and are willing to give or sell to you for a nominal fee, get the phone network/SIM unlocked if necessary (there may or may not be a charge for this - check with the operator), then buy an appropriate SIM pack to use with the handset.
Go for a pre-paid phone with credit on it (meteor prices are good); if the phone is not used much the bill pay option is too expensive. Most operators require a minimum top-up at least every six months.
Check out carefully if your efforts on your grandmother's behalf are viable.
I wanted to help my mother (then in her 80's) with her anxiety around phoning family members (most living abroad, so long codes and hard work dialling if your fingers are all twisted with arthritis and the phone-point is next to the front door in a draughty hall).
Despite long careful instruction she never got used to a cordless push-button phone with the family's numbers all pre-set.
At one stage after a week of not being able to get through to my mother I had to phone an uncle and ask him to go around and check what was happening. We never worked out what had happened but she asked for her old (known) instrument to be reinstalled.
Perhaps lend her a hand-held for a week as a trial to see how things go?