I Wrote a Children's Book and am Thinking of Publishing It

My children's book has been an on-and-off project for quite some time, but I finally sat down and finished the written portion. My relatives keep pestering me to publish it. I haven't illustrated it yet because although I love to doodle, I've never been consistent with my drawings and I would hate for the children to think they were reading about cats at first, only to find at the end of the book they were badgers and horses all along. Here is a little teaser from my book, Titled "It's Raining Cats!"

By morning there were piles of cats
As far as eyes could see
Stuck in every flower pot
And perched in every tree

The Farmer didn’t know what to do
He shouted with alarm
He found a herd of angry cats
Had trampled through his farm

The Baker had lots of bread to sell
Before the cats arrived
Late last night they munched his goods
So he sat down and cried

The Butcher had been robbed as well
His shop in disarray
He shook his fist at the naughty cats
“I’ll surely make you pay!”

Cat hair floated through the air
Carried by the breezes
The quiet, restful little town
Was filled with many sneezes

Dear reader, do you have experience with writing or authoring or publishing or illustrating? Advice and tips would be gladly received.