Re: shoulder pads - was actually curious about them too. The plates on the front and back at least seem to have bolts, while the shoulder pads, I suppose, are held by something under them. You should note though that the uniform is not in fact an up-armoured service uniform, but uses a different tunic, so these parts may be just solidly sown onto the shirt - or be held by huge pads of Velcro, which isn't unheard of in modern uniforms and NASA gear.
So, to answer your question, with Battlestar Galactica "dress blues" as template, the upper part is presumably a double-breasted tunic, with one of the lapels (the outer one) fitted with additional ballistic weave reinforcement (remember, Illusive Man's elaborate suit is bulletproof). There's also probably some form of undershirt down there. So, the whole thing is put on not unlike a button-up shirt, except that there's an overlapping central part - which can be held together using zippers, but more likely more Velcro, because it can't be turned into a projectile or frag. The same is almost certainly applied to Alliance dress blues (which have at least decorative buttons) and probably the utility garb as well. Ditto for Cerberus officer uniforms.