Agree with MarlinG... A well maintained compost pile will heat up but never combust.
Wet hay bales do sometimes catch fire, but they are very packed and full of hay dust which is very flammable and they are not turned or anything. If you want to prevent that in the hay bale, you just have to open it up and toss it around a bit.
But the compost pile full of lots of different ingredients, mixture of brown and green, and kept a little damp, is not an issue.
Congratulations on starting composting! Best thing you can do for your garden! And great for reducing waste. I compost everything... used paper towels, coffee filters, the bags the bird seed comes in, etc.