In my opinion Caversham is lovely and an excellent place to live.
It is a village separated from the rest of Reading by the River Thames but has lovely countryside very close within 5-10 minutes by bike. Yes it is near Reading but Reading has good shopping, a cinema, a theatre etc and if you don't want to go there you don't have to! It is very easy from Caversham to head into Henley or Oxford.
Oakley Road is one of the better roads in the area and should get you into Caversham Primary School which is rated outstanding by Ofstead (but please check with the council on up-to-date catchment areas). This is generally thought of as one of the best primary schools in the area and has huge demand for places.
In terms of secondary there are 2 grammer schools in Reading - Reading School (boys) and Kendrick (Girls). Both of which are rated very highly. I think Reading School came out about 10th in the whole of the UK in the Sunday Times ranking and Kendrick was almost the same. Clearly these are selective schools. Oakley Road is in the catchment of a large Secondary school called Highdown.
Caversham has a range of independent shops, a Waitrose, a really good independent butcher, and several independent places to eat and drink. Oakley Road is near several bus stops that will take you to the Station in only about 10-15 minutes in rush hour or you could walk it in about 20. The train itself only takes about 25 minutes to London Paddington although it does get busy.
You could live further out eg Goring etc but any Journey back to London will take longer and cost more in terms of the ticket price.
As others have said, problem is getting to the motorway across Reading but most people either leave early or late and avoid rush hour if they need to drive. Also relatively easy to get to the M40 from Caversham.
In terms of a fast commute to London, reasonably priced period housing (compared to London), good schools, and a good quality of life, Caversham ticks all the boxes for me. We moved from London and it was the best decision I made.
Only further advice is to just make sure you try to move to Caversham Heights, Caversham Central (close to both bridges or near the main shops) or Emmer Green.
Good luck!