Here is one Zoom vs. Prime Lens Sharpness Comparison on Flickr. The prime lens looks sharper. But of course, this is just one case. As someone points out on the first thread that @mattdm linked "Prime lenses are usually sharper than zooms". (emphasis added)
Perhaps you would be more interested in comparing a zoom lens you already own to a prime lens you are considering. Normally this is where I would refer you to DPReview.com and tell you to check out the lens reviews. Their reviews have a lot of great data on the performance of the lens. Sadly it seems there are a lot of lenses they have not reviewed. So once you know what lens(es) you are considering, you might have to search the forum for posts related to the lens(es) or post your own question.
There are a lot of professional/experienced photographers on these forums that love to use prime lenses, and of course their forum posts will reflect this. Rightfully so. But that doesn't necessarily mean prime lenses are right for you. There are also plenty of newer photographers on these forums. For newer photographers I would recommend using a zoom lens for a while. Good starter zoom ranges are 18-55mm or 18-135mm lenses. Then you can keep an eye on what focal length(s) you use the most by checking the EXIF data in the pictures, or just by paying attention to it as you shoot the photos.
If you want someone to pick for you ... well if you're going to twist my arm then here are my two cents: It seems most people opt for a 50mm for their first prime lens. But if you're more interested in landscape then 24mm would be more suitable. As you said, I personally wouldn't go wider than 18mm unless you're looking for a little distortion/fisheye effect. And you probably already have a zoom lens that will get you down to 18mm if 24mm isn't wide enough. As far as medium telephoto: Well, 50mm counts as medium telephoto unless you are shooting on a full frame camera. (At least according to Pentax.) If you are looking for something a little longer that you can still take portraits with, consider 85mm. But again, I would use a zoom lens to see what these focal lengths look like and what kind of shots you can get with them.