Firekindling training will allow you to more easily start fires and eventually teaches more specialized firestarting techniques that employ materials that can be foraged out in the wild.
You must have an item to start the fire, and currently only firestones are available.
To build a campfire, you must generally be in the wilderness. If in a forest, you can forage for sticks. You will need a number of them made into a bundle. You can drop this bundle and light it. Note: You can also forage for tinder which, when held in a hand, greatly increases your probability of lighting the campfire. Even if you fail to light the campfire, the tinder will be used up. While sticks are only available in the forest, tinder can also be found in the plains.
You can chop split logs which can be added to campfires to increase the amount of time they will burn for.
The campfire command will explain everything you need to know as well.
Other items that can be lit will generally show than when looked at. Some items will burn forever when lit, while others may run out of fuel and either disappear or need to be filled with things like oil. These items include stoves and ovens, lanterns, mine sconces, and even arrows or paper.