When I first got into photography, I only had access to equipment which my father handed down to me which was a Nikon D200 at the time with a 24-70mm f2.8 Nikkor lens. As I progressed and learnt about what lens I should be using for different subjects and scenarios I fell in love with Prime lens. Choosing which lens you use is one of the most important lessons to learn at the beginning, it is just like choosing a weapon for a fight; you don't want to bring a knife to a gun fight! Each lens is good for certain types of shots and it is crucial to learn the basics before you venture out and try breaking some of these rules.
I personally prefer prime lens(non zoom) over zoom lens but I will try to be as subjective as possible in the breakdown of this discussion.
Benefits of zoom lens:
1. Portability - you could easily just carry one zoom lens that acts as your wide angle AND telephoto lens
2. Easier to compose images, you don't need to physically walk forward or backwards to get that ideal shot
3. You don't need to waste time changing lens and potentially missing a shot, this is crucial for sports photographers, wedding photographers and other professional photographers
Benefits of a prime lens:
1. Price - it is generally much cheaper than a zoom lens
2. Quality - less moving parts, less things to break potentially
3. Photo quality - with less glass inside the lens, the optics are also better but only marginally
4. Much larger aperture - Generally speaking prime lens have bigger apertures which are useful for low light and shallow depth of field shots. There are exceptions, e.g. Sigma is now releasing some excellent zoom lenses with big apertures
5. Better training - because you are shooting with one lens at a fixed zoom, you start to develop a sense of how that shot will look in your mind without looking into the viewfinder. This means quicker reaction and higher quality shots
6. Can shoot towards light better than zoom without too much lens flare and chromatic aberration
When I travel I generally pack two lens - my Sigma 24mm or 35mm f1.4 and my Nikkor 85mm f1.4. The wide angle for landscapes and generic photos and the telephoto lens for portraiture and street photography. This is just my preference, more than welcome to contribute with your point of view and preferences.