I've been having this conversation with my Mum this last week. She has an old PAYG phone that works perfectly well but is not a smart phone. My brother had suggested that she got a smartphone so that they can send updates about the family to her without having to go to the trouble of writing emails etc. You can just buy the smartphone outright, and then decide which network provider / tariff you want to sign up to - or you can sign up to a contract.
When deciding whether you want to use PAYG or go onto a monthly deal (which can either be for a fixed period, or monthly if you buy the handset separately), consider how much you are likely to use the phone. I don't make many calls and don't send that many texts, and I am usually within range of wi-fi at home or work, so I don't use much chargeable data. Therefore, Three's 3-2-1 PAYG tariff does fine for me. It's the cheapest of the major networks, at 3p/min calls, 2p/text, and 1p/MB data (£10/GB). I reckon I put about £10 top up on it every 3-4 months perhaps.
However, if you're going to be spending more than about £4 per month then it's worth looking at either of the types of contracts.
Some handsets are very cheap. Mine is a Motorola Moto G - a good little smartphone but doesn't have a great camera. The current version is about £100. I've been very happy with mine and its predecessor.
If you're going to end up changing network provider, consider how much PAYG credit you have left currently. Mum has got about £16 credit, which is not to be sniffed at. I'll suggest that she uses that up before switching to a new provider, or she'll lose it.
Feel free to ask any other questions