I set an unlit campfire on fire?

If you want it in that form, try "I set a campfire alight".

"I lit a campfire" would be more usual, but typically encompasses the entire cultural act (collecting wood, finding a suitable place, tending the fire until it is established, that it is somehow "your" fire, and so on) rather than just the act of setting alight.

So, for example:

"After a long day of hard work, he lit a campfire and made a refreshing cup of tea"

but

"High on amphetamine sulphate, he rampaged through the scout camp, felling flagpoles and setting campfires alight."

You can also "start a campfire."

According to a police report, he started the campfire on Wednesday night and stayed up all night sitting around it.

He started the campfire and she got the coffee pot ready.

To light a campfire may be a way to express what you want to say.

You can "kindle a fire" a phrase which can even be found in some version of the bible. It's little bit archaic, but I've heard it used.

Note the relationship to the word "kindling" for small easily burnable stuff (just one step up from "tinder".