So I have an Argonian character as my first character. I am really wanting to get in on the Imperial/Stormcloak conflict, but since I am an Argonian I can't help but feel I do not fit in anywhere...Argonians are more or less treated like crap in Windhelm by Stormcloaks, and the legion seems a bit too formal for an Argonian...

Does an Argonian have a place on either side? Like I said, I want to get it on this but I may just have to make a Nord character or something, which I don't particularly want to do, but won't spoil the questline if an Argonian makes no sense at all.

Argonians have no problem with formality. The legion would welcome you with open arms.

This is true, although I cannot help but feel out of place in the legion, they should throw a few beast races in there to make it feel more diverse.

I'm fairly sure I've seen beast races in Imperial Legion garrisons in Morrowind.

This is true but keep in mind that Morrowind was set a few hundred years in the past when there was a unified Empire. By the time of Skyrim, the Argonians have their own nation so I'd imagine it would be extremely rare to find an Argonian serving in the Imperial Legion unless he/she came from an Argonian family who have lived outside Black Marsh for generations and are staunch supporters of the Empire.