Morning AnyaParish.
I have two children, a nearly 6 year old girl and a 7 year old boy.
Here is a list of things you might find useful. I have commented as best I can:
Sevenoaks Leisure Centre - www.sencio.org.uk/sevenoaks/about:
Taekwondo - super class, teachers amazing. Both kids really enjoyed it but we cancelled as boy started judo at school.
Charlton Athletics Football (indoor during week). Great, all round and friendly. They also arrange school holiday football coaching as well, and very good value.
Swimming lessons (my boy doesn't do it here, but you can with either private or group lessons available - all very good but contact soon to get a slot).
Sevenoaks School Swimming - www.sevenoaksschool.org/swimming
My boy has his lessons here after school. He also has lessons at his school which doesn't have a pool, but they also bring the kids to swim here at at the leisure centre. The swimming pool is amazing, new and extremely clean. Great location and teachers are excellent.
Sevenoaks Football Club - www.sevenoakstownfc.co.uk/#!:
Saturday mornings. Open to kids aged approx. 4 - 11 Friendly, super skills learning and great opportunity for dads/mums/kids to meet. Contact them for further information.
Hilden Grange Golf - they do Sunday morning classes for the smalls. A great chance to learn how to play and very reasonable.
There is also the Sevenoaks Rugby Club, but we like to keep Sunday free for family.
There is Hollybush Tennis Club. We don't do this but have had excellent reports and friends that do this. tennis.isport.com/tennis-courts/gb/kent/sevenoaks/hollybush-tennis-centre-116875
There are also other options:
Nizels - based in Hildenborough - a private members gym. They do ballet, judo, taekwondo and swimming lessons for little ones.
St Julians - a private members club set in beautiful surroundings. They have tennis courts, open play areas, extremely family friendly and 3 swimming pools. No lessons as such, but well worth a look. Super food and social scene.
I hope the above is helpful. Please feel free to PM me if you need more info. Good luck with your move !
Px