What checks should you perform before setting off to ensure that your bike is safe to ride?
I'm more interesting in mountain bikes rather ride road bikes but I guess that most checks are applicable to both.
I think those are whats needed to make sure your bike is safe to ride. If anything else breaks while you are riding you should be able safely stop and deal with whatever situation arises.
There are some more things you should check every time prior to every ride in extreme weather conditions, but I think those three are universal to all bicycles.
It's good to check out other areas from time to time or even every time, but I think those are the "mission critical" things to check.
My pre-ride check is to sit down with my bike and do some maintenance on it the night before a big ride. Clean and lube and adjust and tighten. Just go over everything. Think about what noises it has been making, whether the derailleur needs adjustment, are the brakes too firm or too soft, and so on. The more time you spend with your bike, the happier you and it will be. As you work on one thing, you might notice something else.
Another thing that helps is to do this with a friend. My friends will point out things about my bike that I never noticed before.
Of course, if you don't have all that much time, check your brakes, check your chain condition, check your quick releases. As long as those are good and you have a spare tube, I don't see what could stop ya!
I check all the fasteners on the bike and my shoes. I've had a cleat come loose during a ride, as well as a cable come out because of a loose connection. These things work loose over time, best to torque properly before each ride (or once a week once they start to take a set and stay tight). Invest in a decent Snap-On (or similar) torque screwdriver, with bits to fit your bike.