So there are only really 4 mobile networks in the UK - EE, Vodafone, 3UK and O2, so you need to figure out which one works best for your exact location.
All the other networks (e.g. Tesco Mobile, Giff Gaff, Lebara etc…) actually use the mobile signal from one of these 4 operators so coverage won’t be much different. Virgin uses EE base-stations for example, so rule that one out if it’s already not working for you!
Next I would recommend looking at the online coverage checkers, though they are part marketing tools, so buy a pay-as-you go SIM and see if it works for you before deciding if you want to buy either a new phone or get a contract. While you can buy phones outright, that can be expensive up front and so deals to get a phone a part of a contract may be a better option. If you do enter into a new contract, be mindful of the cooling-off period so you can cancel if you find coverage isn’t what you need.
Voda and O2 both claim good coverage for 2/3/4G across Sydenham:
3UK looks a bit more patchy:
You can buy Pay As You Go SIM cards from newsagents, usually for something like £1 or £2, but you will have to add credit (usually £5 or £10 minimum) which can either be done at the newsagent (if it is a paypoint), via the phone or even at cash machines. You can also buy SIMs on Amazon, with or without credit.
Tablet vs phone is something of a personal preference. Some phones are quite large these days and almost like mini-tablets. Probably best to look in a shop at some of them and see what you think.