General Observations

As your adventures get longer, and farther away from bike shops, the more you will want to learn how to maintain and repair your equipment.
Parts you take may save you time, but slow you down.
Parts are easy to justify taking if they are either small, or hard to find, or if you have a schedule without wiggle room. This includes spare screws for rack, water bottle cages, and cleats, and an extra tent stake.

Short-term trips (less than two weeks)

Thumb-test your tire pressure every morning. Riding 10-15 pounds low is a drag.
Rain and puddles call for frequent lubing of your chain. If you run out, chain saw oil or car oil will work - just remember to wipe off everything that doesn't penetrate. You know you waited too long when you hear your chain squeek, or you get chain suck or skipping from the links not being flexible.
Know how to use the thumb-nuts to adjust your shift and brake cables when they stretch.

Moderate trips (two weeks to two months)

Once a week, squeeze-test pairs of spokes, looking for those that are loose or broken. A loose spoke puts extra stress on the other spokes.
With the bike upside down, spin each wheel and look for wobble near the brake pads. Adjust spokes with a spoke wrench to minimize the wobble.
With the bike upside down, slowly turn the crank backwards and watch the chain at the rear deraileur. If it jumps, a link needs cleaning and lubing. Without timely service, you will discover the problem on a steep hill as you experience chain suck and everything comes to a sudden stop. Just hope there is no traffic close behind.

Long trips (over two months)

From time to time, check each tire for cuts and imbedded objects. Don't remove anything if you don't have time to make a repair.
Learn how to replace the bearings in your wheels. Take more than enough bearings, of the right size, and a small container of grease.

Sudden small things

Sudden large things

Learn how to repair a cut tire with a dollar bill.
If your tire is damaged at the bead, you can sew it together with a large needle and dental floss.
If you blow your last tube, and the hole is too big to patch, cut the tube at the hole, tie knots in each end, and stuff leaves and grass in the tire where the ends don't meet. It will be bumpy, but is better than walking, or carrying your load.
If you break a shift cable and don't have a spare, force the deraileur to stay on a cog or ring that is comfortable for riding. You may have a 3-speed bike, but should be able to have a gear low enough to get started on level ground.
If you break several spokes near each other, you can move spokes from other parts of the wheel, making it easier to reduce wheel wobble. Take spokes in pairs from one side to make pairs on the other.
If you break a spoke, or several, and can't get the wheel true enough with your spoke wrench to clear the brakes, unhook the brakes and ride carefully.