But they never showed even once who the enemy they were fighting against, was it the russian army? Or sator's private army

You don't get a close up shot, but in the opening strike by Red team, you see a bunch of soldiers in white/grey camo shooting at Red team, that's the enemy. That is Sator's private army

(can he even afford it)?

After Sator's $9 Million+ vacation to Vietnam, I think he can afford his private army. Also what does money/debt mean to someone who thinks he's going to take the world with him in a few hours?

Or were the red team and blue team shooting and killing each other to make some time-travel magic happen?

No, how would that help Red and Blue teams?

Nolan did a bad job here, he should have made every different group wear a different colour costume so the audience would know what is going on.

Red and Blue teams wear green, the enemy wears the white/grey camo. Especially visible when we see Volkov, as he's wearing the same outfit.

The battle between Red/Blue teams and Sator's army isn't the focus, they're the distraction for splinter unit who is the story's main focus