Of course, I am not an expert, and every computer/setup is different. This is how I managed to finally get it to work on my computer. I use Photoshop Elements and Safari.
Start by making your animated GIF. To do this, save each frame in a new layer in Photoshop. Make sure your image is low resolution.
Go file-save as GIF. Select "layers as frames."
Make sure you select animation, loop, and the timer that you want. Preview to make sure you are happy with it. (I discovered that the time won't change from .2 in my version of Photoshop Elements unless I deselect animation and loop after I change the timer, then reselect it quick. I'm not sure why that does that.)
Load your GIF onto an online photo sharing website. There are a lot of different ones you can use, I use Photobucket.
Once uploaded, select the "share" feature and get the HTML code for sharing your image.
Go to "edit" on the blogger post you'd like to add the GIF to.
In the upper left-hand corner of your navigation bar, select "HTML." Paste the GIF code that you grabbed from your photo share website where you want the GIF to appear.
If you go back to "Compose" from "HTML," you can easily edit things like positioning, etc... If you don't want your GIF leading back to the Photobucket website that you pulled it from, remove or change the first web address in the code.
I hope this helps some people, though if you are running completely different programs than I am, there are tons of tutorials online that you can check out. Good luck!