The shifter for the front derailleur is "soft" in my vintage bike So I bought a used vintage bike. I just love the bike and it's awesome. However it came with one problem: gears problems. First of all I have very little experience with bike, especially gears (currently watching youtube videos to learn more).

The problem that I have in the bike is three folds:

1- The shifter for the front gear is soft, so there is absolutely no tension when you move it. So if you move it, nothing actually changes in the front gear at all. My question regarding this problem: could it be the shifter itself that is broken or could there be something else wrong? (i.e. like cable problem, or front gear problem itself). I'm trying to troubleshoot the problem.

2- I have never seen such shifters ever before, so if the switchers themselves are broken, are there still old school switchers like these? If so, do they have specific names so I can look them up online? Because when I search for "shifters", I only get the new popular ones!

3- The last problem has to do with the rear gears, the switcher doesn't switch to all speeds and I have no idea what's the problem.

Here are pics for my cool bike :D (so happy about it):
At first glance I see the problem you are having .

Those are NOT friction shifters . They are positron shifters .

Research Shimano Positron Shifters for a more definitive explanation  

This was Shimano's earlier attempt at an indexed gear system.  Also known as "Positive Pre-Select" or PPS.

The main differences are that the indexing is in the deraileur rather than in the shifter on the handlebar.  As a result, the bowden cable is a full housing and has a thicker solid inner rather than a woven stranded cable.

Upshot - you can't just fit any old cables to the rear mech.  However its possible the front deraileur is just a standard non-indexed friction shifter.