I'd like to buy a better mobile telephone, one which can show email and the odd web page. Can they use the wifi router when at home? It would be good to know what is available before approaching a shop.
I'm guessing you mean a smart phone and there are so many available and so many different payment options and tariffs it would be good to have some idea before you go into a shop and they bamboozle you with sales pitch.
Yes they use your home wifi router for the internet at home.
Most phones now are smart phones (except ones specifically designed to be very basic). They tend to have large flat glass screens. If you ask in the sop they will guide you to the smart phones.
I would recommend that you go to somewhere like a Tesco Phone shop on a quiet, mid-week day and ask them for advice. I don't shop at Tesco but the phone shop in our local one is excellent.