Gomshall station is nearest to Peaslake, though less frequent trains there.
Peaslake is the bike Mecca part, though there’s a vast amount more across all the hills. If you’re after singletrack, around Pitch and Holmbury hills is easiest to find them without a guide though still requires exploring and easy to get lost if you don’t know the place.
Can ask around or in the little shop (Pedal and Spoke) for pointers, though not sure when the shop is open during week days.
Bridleways stuff is easy, tonnes of it. Plenty of guides online or in the shops will give routes or check out stuff like Open Cycle Map. You might be better looking more around the North Downs Way or round Leith Hill for that kind of riding. Singletrack with roots and steeps, gaps and jumps, are more around Peaslake way. Though there’s loads of hidden gems elsewhere.