Once upon a time zoom lenses probably were inferior to prime lenses in terms of sharpness. This is not the case anymore. If you buy a good quality zoom lens it will compete very nicely with a good quality prime lens in terms of sharpness. Most of the criticisms of zooms' sharpness comes from pre 1990s experiences. I have a late 90s Pentax 28-200 zoom that I've blown up Tmax 100 negs to 16x20, and they are amazingly sharp. There is going to be a sweet spot somewhere in the range of focal lengths that is the absolute sharpest, but even at the ends a quality zoom will be very sharp. Once the engineers and factories got access to powerful computers it became very easy to figure out the problems and correct them.
There are other issues than sharpness though.
Lens flare: every glass to air surface among the lens elements can reduce color saturation a bit (a tiny bit), and increase lens flare potential. Zooms have many more elements than prime lenses, so they have the potential for more flare.
Speed for your buck: You can find fast (f/2 or better) primes quite reasonably priced. Zooms faster than f/4 or f/5.6 though usually cost $1000+. This can be a big deal if you are into available light photography. Also, most lenses are sharpest 2 or 3 stops from wide open, so if this is where you like to shoot it turns your f/2 50mm into an f/4, and your f/4 28-105 into an f/8, which may mean shooting ISO 400 instead of ISO 100.
Learning curve: I think that zooms get in the way of learning about focal length. Most beginners spot the image, and then crop with the zoom, not paying attention to the focal length they are using. They never get a feel for how focal length affects the image. The right way is to spot the image, decide on a focal length, set the focal length, and then crop by moving closer or farther away from the subject.
If I could only have 2 lenses for my 35mm SLRs I would pick a 28-200 (or even 28-300) zoom and a fast 50mm prime.
I don't know exactly when he wrote this, but here is a quote from Ansel Adams "The Camera" (<--THE BOOK on how film cameras operate, and would probably even teach a thing or two to digital only folks).
"In earlier years some lenses were definately superior to others; their images proclaimed their differences. I have a Voightlander 12" process lens that is suberb, despite it's age of more than 70 years, and I have done some of my best work with Zeiss Protars, nearly 40 years old. More recently, the computer has moved into the industry, and practically all lenses made within the last decade or two are excellent-often more precise than even the most exactingpractical photographer requires. They are, in fact, usually better in their capacity for preserving fine detail than our standard contemporary films and papers can record."