You have three main options for a smartphone, iPhone, Android or Windows Phone. I also recommend a "Pay-As-You-Go" plan on the assumption use will be limited and you have WiFi. When you do buy the device do not purchase one from stores such as Argos. Instead go to an EE, O2, Vodafone or 3 store as staff will be able to help you.
An iPhone is very simple to use, however Apple's devices are expensive and have fragile screens which crack easily. I do not recommend an iPhone on cost grounds, however Apple's devices have the most "apps" (programs that add features to the device other than calls/texts).
Android devices come in many forms. The main company making Androids is Samsung. The Samsung Galaxy devices are easy to use, and have a large number of apps. Some cheaper Android models exist however usually take longer to load/turn on. I would recommend a Galaxy S6 or HTC One device - as these are popular, easy to use devices.
Windows Phone devices are the easiest (in my opinion) to use and cheap, although the range of apps is limited. Windows Phones have an interface using large boxes, which are easy to see and use. I would recommend the Microsoft Lumia 640 if you were to chose a Windows Phone device.