Public Transportation web sights in Great Britain

For trains in general, use www.nationalrail.co.uk

If you want to include buses and other transportation as well, use www.traveline.info

None of the above, if you mean by Public Transportation Trains and Buses and other modes, like ferries. Rome2Rio is useless, and can be discarded immediately.

For integrated public transport information which gives all modes use Traveline- https://www.traveline.info/ for England.

It has spin off sites for Wales and for Scotland- https://www.traveline.cymru/ and https://www.travelinescotland.com/

Those two sites for the devolved nations also give fares information- in Wales on the whole it seems to be pretty good if not stellar, in Scotland the fares bit is woeful but the travel data is good.

Google Maps with its directions by public transport is a good starting point. Depends if you want long-distance public transport or local.

For major cities, City Mapper App can't be beat. I even use it at home here in Seattle. It will list all tube, bus, walking options.

Train: National Rail https://www.nationalrail.co.uk it will link you to regional train companies.
Bus: National Express https://www.nationalexpress.com/en

Local bus companies. You will need to be aware of what companies serve the local area you are exploring.

Within the greater London area, https://tfl.gov.uk/

Advice on trains:>https://www.seat61.com/train-travel-in-britain.htm

I have to say, ALL of these posts are full of good, useful information you can use. That's just me of course...

Thank you, thank you for your responses!! We have always rented a car BUT sadly we are now 74 and not allowed to rent. (Probably a good idea) but we love trains and buses!!

One more question:What about Ireland???