Is there an online service or program I can download that will allow me to compress animated gif files?
I know that. 
But I cannot recommend a ‘specific one’ as such because I prefer PNG-8 and have little purpose for the inferior GIF format, let alone animated GIF.
If the images were PNG in the first place they’d probably be smaller. So that is a suggestion convert the individual frames to PNG before doing the animation and layering processes it may just make them slightly-smaller when you resave the PNG back to GIF for the animation part.
I have seen several handfuls of online ‘Animated GIF optimisers’ some free and some not they’ll all work but which one can be considered ‘really good’ is another matter? Sorry, but I cannot really offer much else with regards to a recommending a specific application (URL) other than trial-and-error.
LoL - that’s the point of this thread, I’m looking for sites or a program that will do this…
Yes, there will be you could do a search on the internet. It also depends upon which Image Editing program you use to make the Animation as to how much it compresses the GIF in the first place some are better than others.