What are the advantages and disadvantages of both client-side and server-side programming?
Your question is sort of like asking what are the advantages of buying a car with a steering wheel vs. one with breaks -- it is rare indeed not to have both.
It is possible to have a "client" app or, more accurately, a desktop app without a server backend -- although that is becoming more and more rare -- but really, without some kind of client, a server isn't much good.
If you are wondering what is better to learn: that is really a matter of personal preference. For example, I prefer server programming because it makes more sense to me. On the other hand, client-side programming can provide much more immediate gratification as you see your programming immediately manifest in a visible GUI. For me, it's just frustrating.
So take your pick. Or even better, do both.