It seems like it's free for all inside on the grounds. You can cycle on the roads and sometimes park next to the entrance to the sub-temples if there is space (not all of them have that). But don't leave your bike in the way, cars are also using the roads.
I saw people parking behind the south gate, but there is also dedicated parking outside. To be safe and avoid trouble, the dedicated parking is always the best option.
Dedicated bicycle parking for visitors is next to the southeast car park. It's on the side of the Hanazono Kaikan hotel (operated by Myoshin-ji). It's also used for motorbikes. If you can't find it, the staff at the car park will be happy to point you in the right direction.
There are signs to walk your bicycle on the way there, so please follow them.
The complex is huge, and while the main grounds are free to wander, most of the sub-temples are closed to the public or private residences. Only a few, such as Taizō-in and Keishun-in, are open year-round for an admission fee. You must check in advance for the current opening status, hours, and special seasonal openings of any specific sub-temple you wish to enter.