Reading branch of Cycling UK (formerly the CTC) organising social rides from Dinton Pastures and Caversham. Road and off road at leisurely pace of 10-12mph on Wednesdays and Sundays. More details here https://www.readingctc.co.uk/
For retired people, there are cycling groups in the Wokingham U3A riding from Cantley Park and Finchampstead on Monday mornings. Loads of other interest groups to get involved in too! More details here https://www.wokinghamu3a.org.uk/groups
Breeze is a women's only group linked to British Cycling organising free fun rides in the area. More details here https://www.facebook.com/BreezeNetworkBracknellForestHartWokingham
FLAB (Fat Lad At the Back) are a clothing brand that created a group for Social Rides. More details here https://letsride.co.uk/groups/flab-social-rides-berkshire
Sprockets is a British Cycling Go-Ride Club for 3 – 18 year olds, running sessions on traffic free sites to improve cycling skills, confidence and ability. It is an ideal follow-on from Bikeability Learn to Ride sessions. More details here Sprockets Cycle Club website.
Maidenhead Cycle Hub are a bike recycling workshop but they also organise family and beginner social rides starting from Braywick and Ockwells parks in Maidenhead. More details here https://cyclehub.org.uk/organised-rides/
There are a few groups on meetup.com organising casual rides. Phoenix run Wednesday and Saturday rides around Camberley, Wokingham, Farnborough & Fleet. Look them up on meetup.com. Search for other groups as well using 'bikes' or 'cycling'.
The British Cycling website has a good event search facility to find all types of cycling events from sportives to recreation rides. Look here https://www.britishcycling.org.uk/events?search_type=upcomingevents and set your search parameters.
If you're looking for purely mountain bike groups Swinley Bike Hub are running Gravel and MTB led rides on the forest trails. Berks on Bikes are a local MTB club who also ride regularly in the area - https://bobmtb.com/
Similar club to WCC with rides starting from Shinfield. Similar speed groups to Wokingham with some fast chain gangs as well. Details here https://swallowfieldvelo.club/
Reading CC is a more 'racing' based club than WCC, so if you are interested in getting into racing its a good place to start. They also do more social rides starting from Reading Market Place each Sunday. More details here https://readingcyclingclub.org/
The BIA Hub in Twyford run a women's only club with rides from the Hub in Twyford. Very supportive with rides at all levels and great club kit. More details here https://biacycling.com/club
Similar club to WCC with rides starting at Braywick Park in Maidenhead. Organises lots of Time Trials. Details here https://www.maidenheadcc.org.uk/
If you're looking to mix your cycling with running and swimming TVT are a big club running rides from Shinfield. More details here https://www.thamesvalleytri.com/
Another similar club to WCC with lots of social rides on Wednesday, Saturday and Sunday starting in Farnborough. Time Trials every week in the season. Details here https://www.fccc.org.uk/
Just into Hampshire, Hook Velo is a membership free group organising social rides of different abilitities from the Community Centre in Hook. Details here https://hookcyclingclub.co.uk/
Racing focussed club organising Saturday and Sunday rides from Henley Rugby Club and the Market Place. Typically 16-19mph pace. Sponsored by the Athlete Service shop in Henley. Details here https://gshenley.com/
Similar to WCC with rides at all levels starting from the Sports Club in Marlow on Saturdays. Organises the very popular Red Kite Sportive each year. Details here https://marlowriders.teamapp.com/?_webpage=v1