Do I need to convert 60fps to 30fps so it will be reproducible? I just bought a GoPro Hero 3 Black Edition.  I have been testing out two modes: 1080p at 30fps and 720p at 60fps.

I have noticed that when reproducing the videos with 720p at 60fps they are not really smooth, they seem a little bit weird (I have used VLC to reproduce them). So, a friend of mine suggested me to convert them to 30fps and then they will be slow motion, and later on speed them up 50% with Adobe Premier.  After doing that, they look better.

I am confused. Why does this happen? Why do I need to low them down to 30fps and then speed them up up to 50%? Wouldn't that mean that we are back to 60fps?

Thanks
You're not back to 60fps, you're still at 30fps but with every other frame discarded (or blended, or interlaced, depending on how the speedup is accomplished). 

When you discard every other frame, you're trading off temporal resolution. If you interlace, you're trading off spatial resolution.

Either way, if this appears smoother, something's wrong with the way the 60fps is being presented. Your playback system may not be able to keep up with the data rate or frame rate. If 1080/30 looks OK, then it's probably not a data rate issue.