Parker needs to have a strong knowledge base focused in chemistry, biology and engineering. He needs to be a math wiz, with a keen grasp of physics and geometry, and he needs to be proficient with computers and laboratory equipment. These combined attributes are what allow Peter Parker to be Spider-man.
Throughout his history he's had to build things like radio transmitters and em field generators (such as the device that disrupted the Vulture's anti gravity technology) and he's had to alter or devise serums (such as when he's reversed the Lizard's transformation). His understanding of physics and math allow him to more effectively predict and control his motions when web swinging, and using his spider agility. Parker is not the 'average guy' who just happened to gain super powers, he's someone who spent his entire life focusing on intellectual pursuits and education rather than physical development, yet after the spider bite he immediately develops super human physical ability over night. Spider-man's powers are obviously useful, but they wouldn't be nearly as useful with out Parker's mind behind them.
The first and most obvious testament to this fact is the web shooter. The invention of the web shooter needs to be an impressive and complicated feat that acts as a first demonstration of the range in Parker's knowledge. The web shooter itself is a feat of engineering and the web fluid is a feat of chemistry, but I think there's also an opportunity here to demonstrate a knowledge of genetics and biology. This can be done by having Parker engineer a transgenic bacteria to generate one of the ingredients in his web formula. Not only would this require an understanding of genetics and biology, but it would provide an explanation for why Parker is seemingly the only one with access to the web formula.
A scientifically minded person like Peter -upon getting super powers- would undoubtedly want to examine himself (and spiders) as a way of better understanding his transformation. Upon sequencing and studying his genome, he should discover a mutated gene responsible for producing a silk protein similar to Spider web. Parker can isolate this gene and splice it into a strain of bacteria and use this bacteria to generate a unique silk protein which he uses to complete his web formula.
Scientists are already engineering transgenic life (goats and bacteria for example) as a way of producing spider silk, because spiders themselves are impractical to farm. The ASM movie was silly in this regard for having a futuristic company like Oscorp farm actual spiders as a way to produce web, when scientists in the real world are already engineering spider goats to accomplish that same task, which ironically seems more like a concept from a comic book.
So once again, for me the ideal method through which Parker generates his webbing would be to cultivate a bacteria that has a unique silk producing gene taken from Parker's own blood. Not only is it more of an accomplishment on Parker's part, demonstrating an understanding of gene splicing, but it's actually more realistic than buying web from oscorp which is generated by actual spiders.
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