I'm beginning to drift from the OP's point, but it's probably worth understanding what (land) public transport options are available in Great Britain. I think it's comprehensive to list these as
- taxis (to include hackney carriages, private hire cars/minicabs/Uber and so on)
- buses and trams
- coaches
- trains, including the London Underground, Tyneside Metro and Glasgow Underground
The first thing to note is that shared taxis don't appear on this list: by all means arrange to share a taxi with friends before calling the car, but don't expect to be able to join one that's already occupied by people you don't know.
Otherwise, taking them in turn
- taxis are probably the most expensive option so not generally suitable for long (> 15km?) journeys. However they will take you from door to door. All the other options will involve you getting to and from the bus stop/station.
- buses and trams run on fixed routes and generally to regular timetables (although note that 'regular' may not necessarily mean 'frequent'). They are generally quite cheap (there's currently a £2 maximum bus fare but this may change later this year so (e.g.) students starting in September 2023 may not be able to rely on this) but slow. Generally these are suitable for short journeys in your town or city or to your nearest town. For most buses you pay the driver as you get on the bus. Someone with more knowledge should be able to explain what happens on buses in London and the various tram systems.
- coaches provide long distance transport and are generally cheap, but with a few exceptions (the Oxford Tube to/from London, airport services. Anything else?) don't run very often. I think that in practice journeys need to be booked in advance
- trains provide both local transport in competition* with buses and trams, and long distance transport. Local journeys operate to a timetable with services somewhere between every few minutes and every couple of hours - check the timetable first to find out what sort of service you can expect. Long distance services operate to a timetable of somewhere between a couple of trains per hour and two or three a day - again check the timetable first. Practically speaking, you should always make sure that you have a ticket before getting on the train. For local journeys, buy this from a ticket machine, the ticket office or on an app on your phone. For long-distance journeys, you can buy a ticket when you arrive at the station (just like for local journeys) but it can be much cheaper to buy your ticket in advance although that will (probably) restrict you to taking the specific train shown on your ticket. Note that some tickets have to be issued on paper, so even if you have bought it online check whether you are meant to collect the paper ticket from the booking office or a ticket machine before you get on the train. There may be better ticketing options as well (particularly on the London Underground) but I leave it to people with more knowledge to explain those.
In terms of paying your fare, my impression is that for practical purposes UK-issued debit and credit cards (Mastercard/Visa) are widely accepted (although not necessarily in all taxis). Cards issued abroad may be more of a challenge as may be AmEx, Diners Club and so on. Cash is generally accepted but as far as I know there's no legal requirement for anyone to accept it: also at one end of the scale trying to pay for something costing several pounds in 'small change' (coins) will not go down well, and £20 and £50 banknotes are not always acceptable (historically there was a risk that they might be counterfeit: these days whoever you pay may not have enough change to give back to you if you have paid a £5 fare with a £50 note).
As people have already given, there's also far more detail to go into. But I think this is a bare outline of how Great Britain public transport works.
*Yes. I know it should be in co-operation. But it isn't, so let's not confuse the advice of what actually happens with what we would like to happen.
