My opinions only - and I never do bus because of railways in my blood....
In most cases if you want bus it would be National Express. Megabus if they serve your route and you don't mind being a sardine.
Yes, National Rail Enquiries is the place to look...
I was going to repeat myself each time, but I shall say this only once - several of these depend on WHERE in London you will be. In some cases different lines make more or less sense depending on where you are starting from. So if it says "depends" below, that's why.
London to Oxford day trip - TRAIN or BUS? Depends. Paddington or Marylebone (GWR or Chiltern) by train, Oxford station is a few minutes' walk into the city. Where do you want to go? Coach (several brands) from Victoria Coach Station to Gloucester Green in Oxford, can also be caught at other parts of London. Consider London Walks Oxford tour if you want a walking tour.
London to Windsor day trip - TRAIN or BUS? Which train Depends. Fastest cheapest train Paddington to Windsor & Eton Central (GWR)(quick and easy change to the branch line at Slough - rhymes with cow) which drops you across the street from the Castle. Slower and more expensive no changes from Waterloo to Windsor & Eton Riverside (SWR), and then a walk up the hill.
(travelling with luggage) London (staying near Blackfriars station) to Bournemouth - TRAIN - yes. Paddington, change Reading if you are nearby, Waterloo to Bournemouth straight through no change (on SWR). In your case I'd probably taxi to Waterloo (even though I only take taxis when somebody else is paying!!). It will depend on how much luggage, which side of the river you are on, and if you'd rather take the Tube or a bus. There are buses on both ends of Blackfriars Bridge which go to Waterloo or tubes.
Bournemouth to Salisbury day trip - BUS? I'd pick train (SWR). About an hour and a quarter, one easy change in Southampton Central of about 9 minutes. Easy. Dunno about bus. Car takes about 45 minutes so bus must be slower, which means not much in it I suppose.
(travelling with luggage) Bournemouth to Bath - TRAIN. Yes. about 2:15 with an easy 12 minute change at Southampton Central. (SWR to GWR). Don't be tempted to change at Reading, it involves backtracking and more money and more time.
Bath to Cardiff day trip - TRAIN. - Yes Train takes just over an hour (GWR)
Bath to Bristol day trip - TRAIN or BUS? Depends. Both fast and frequent - the two towns are close. Probably a local bus would be slower. Whichever suits where you are and where you want to go. Ask the hotel or landlady.
(travelling with luggage) Bath to London - TRAIN. Yes (GWR). Fast and frequent, into Paddington.
London to Cambridge day trip - TRAIN? Depends. If you are near The City (the Square Mile) Thameslink goes straight there, in about an hour measuring from St Pancras International. If you are in the East End, Greater Anglia trains will get you there in twice the time and less money from London Liverpool Street. There are also Great Northern trains from Kings Cross, about an hour, about the same price as Thameslink.
I hope some of this makes sense, and some of it helps....
Good luck, it sounds like a fun trip.
If you are in Bournemouth consider looking for Boy Scouts and Red Squirrels on Brownsea Island.
