Simple rules; I have violated most of them. (It’s good to have some sense of adventure.)
Know where food and water are available on your route. Carry food and plenty of water unless you know they are available.
Have flexible layers of clothing. Use sunscreen. If you are heading out to the serious boonies, think about how well what you are wearing and the food you have would work for you if you got stuck in the woods overnight.
Don’t be afraid to spoil the aesthetic perfection of your bike by putting a functional bag on it.
Carry a map. In the Forest, carry two maps and some form of GPS with a legible map.
Your cell phone won’t work way out in the boonies. It might work next to a tree or rock with CELL painted on it.
Don’t take an unsafe or poorly maintained bike out on an adventure.
Carry appropriate tools and supplies including a tube, patch kit, and pump. Know how to fix a flat tire, and know how the rest of your bike works.
Tell people who like you where you are going and when you expect to be back–leave them a map if they don’t know the area of your ride.
Know your level of conditioning and your limits, and choose your routes appropriately. Look at the elevation profile for your route. In the Forest, ask about road closures at the Ranger Station (but not on the weekend). If you haven’t ridden your planned route, could you finish the ride if the road were impassable 3/4 of the way in?
Riding a bicycle on closed roads is legal in the National Forests, but if you crash or your bike breaks down, nobody will be driving along the road to help you.