I bought a home in Whiterun, and I notice there is a fire pit.

How can I make a fire?

And where can I find the wood if needed?

Usually near any wood pile there is a stump. Activate that stump when you have a "wood cutters axe" in your inventory.
I know there is one in Riften by the smithing stations, but they are all over the place as well.
After you chop wood, there will be firewood in your inventory.

You can also buy upgrades for your house, one is a "kitchen" which should give you a permanant fire, not sure haven't bought it yet. You should have gotten a book when you bought the house telling about the different decorations you can buy from the same guy you bought the house from.

Good luck