Ignoring the bikes aspect…
Whitsand / Tregantle beach (low tide only) – nice sandy beach, long walk down from the lay-by at the top where you can park though. Good for gentle body boarding if you’ve got kit. Generally if you want to surf and hire kit you need to head to the north coast e.g. Watergate.
Downderry to your west is a bit of a dump, Seaton is ok, but better would be to go to Looe. Really nice walk along the coastpath to Polperro via Talland Bay where you can reward the kids with fish & chips etc.
Going east, Rame Head is nice. Mount Edgcumbe country park is worth a wander – spot the deer, good view of the dockyard if any of you are into navy boats, subs etc. Kingsand and Cawsand are scenic little villages – park there and walk along to Rame Head.
Further west, Fowey is very nice, quite upmarket. Good coast all round basically. Skip Par/St Austell/Carlyon Bay etc, it’s a dump. Also skip Mevagissey. Pentewan has a good beach.
The Eden Project is definitely worth seeing and there’s a fair bit for the kids to do but it isn’t cheap.
You can get the chain ferry from Torpoint over to Plymouth – the Hoe and Barbican are good for a wander, plenty of nice galleries, cafes and ice-cream shops etc. National Marine Aquarium is a bit pricey but would be good for the kids. There’s also a Vue multiplex behind the aquarium but it’s a bit tired -good wet weather option.
GoKarting and Zip Wire here: http://www.adrenalinquarry.co.uk/