First off, you need a Squiby that you adopted like mine below.
Next, post your pet and click it. It will take you to a page that looks like this:
Stay on that page^ Then hit the refresh button, or the F5 button on the top of your keyboard. For each time for hit the refresh button, your pet will level up.
Hit the F5 button as fast as you can for best results (you can get your pet to level 1000 in a minute or two using this method)!