Doing incremental upgrades is a very wise move - not only does it spread the costs, but it also gives you a much better idea of the effect of each change you make.
For mountain driving (and, in fact, for any situation), the first things I would look at are the brakes and tyres. The tyres are vital, as they're the only things keeping you on the road! The particular choice will depend on your location, the size of tyre you're using, and the weather you're expecting to drive in, but whatever you choose, make sure it's a good brand. Don't go for low-profile tyres - they might be good for smooth highways, but they'll give you an awful ride on potholed mountain roads...
The first upgrade with the brakes will be the pads. Uprated performance pads are usually not much more than standard - but buy 'fast road' pads, not 'race' ones - the latter will be designed to work very well when hot, but can be poor when cold. Vented or drilled/grooved discs (rotors) can then follow.
Moving on to suspension, on an older car I'd always suggest poly bushes, but on a nearly-new car they won't need replacing for years. Stiffer shock absorbers will tighten up the handling, but again the OEM ones ought ot be good for a few years yet! Many aftermarket shocks are adjustable, so you can tweak them to suit your driving style.
The engine is always the last thing to look at - and on many modern engines a chip or remap is the only feasible upgrade without major work. regular maintenance to keep it in tip-top condition, and high quality parts (e.g. spark plugs) are of course vital. If you're doing a lot of high-altitude driving then a more free-flowing air filter might make a little difference.