I think you should get a contract. This gives you a phone and a SIM with a number of free calls and free texts a month and a certain amount of data (for connecting to the internet).
If you go into a phone shop they can suggest a contract and when you sign up they give you a free phone (usually) and you pay so much each month (£15 or £20 say). Dont let them talk you in to paying much more as you dont need a great phone or a lot of internet data
The contract will be for 2 years and at the end of that time you will have paid for your phone and you can either get another up to date phone (usually for free) or you can "keep" the phone and just pay for the SIM (say £10 a month).
I would get a phone that runs Android (this is the software that controls the phone) as millions of people use Android so it is very easy to get help.
Note you don't need to download your photos from the phone to your PC to put on Facebook, you can load them directly on to Facebook from your phone.
Also Whatsapp is is easy to use and is free.
There are plenty of inexpensive phones, the shop will show you the choice at the "low end". However get a decent size screen as modern smartphones can show a lot of data so it needs to be a decent size.
Nokia has just bought out some decent Android phones that wont be too expensive, but to be honest there are loads of others.
NOTE: You can BUY the phone (sim free) from Argos for example (or from a phone shop) and then JUST get a sim monthly contract. So the phone is yours straight away and you just pay so much a month for the SIM (£10 or so).