Once you pick what side you want, is there any reason to not go and kill members of the opposite faction? I'm going NCR and I'm still technically neutral with the Legion. This is my first time playing through Fallout NV so I thought I would do some asking before I started my mass murder spree.

I would avoid it, otherwise you will be dealing with the assassin squads the legion sends. Have you talked to Caesar or Mr. House yet, just out of interest

Well, you don't "have to", but I think it makes more sense for an NCR person to kill legionnaires than it does for a Legion person to kill NCR.
Why?
Cause NCR has more quests.
And NCR affiliated quests allows you to screw them over.
For example, turning the Kings and NCR against one another for Legion, that works.
Helping Vulpes Inculta with that informant in Vault 21 for NCR, that doesn't work.

So if you're helping NCR, sure, go nuts, kill legionnaires.
But if it's your first time playing New Vegas I suggest you go talk to Legion before you kill any of them.
Talk to Caesar, Vulpes, Lucius, Aurelius, Anthony, Ulysses, Graham and Dead Sea. After that, go nuts.
But at least give them the chance to explain themselves.

Thanks! I haven't talked to House or Caesar yet. I just jumped on the NCR bandwagon at first chance.

personally I Don't find the NCR the best choice, just remember talk to Caesar and House before deciding you want to stay on that bandwagon

Yeah... Don't buy into the whole NCR propaganda...
They have their good points, but just remember they also have their bad points.