Especially those kind of glossy but at the same time vector-looking red parts, that have a really nice style. I guess it can be done by using different layer styles. I tried but my result doesn't look like it. I made different solids with different gradient overlays and masked out some parts to put it in the right shape.
Can anybody give some advise? or are there any tutorials out there that can help me.
Don't make it too complicated - I wouldn't get into layer styles or anything like that. I would start with the base, rounded, aperture-looking shapes and then start building up the gradients on top of that. For example you make the big, round, red thingy; then you duplicate it; apply a gradient from white to black; do some masking and finally set it to the "Add" or "Screen" blending mode.
Now repeat this a million times! (there is a reason it's a template...)