How can you make harmonics on a string? For an oscillating string that is clamped at both ends (I am thinking of a guitar string specifically) there will be a standing wave with specific nodes and anti-nodes at defined $x$ positions. 
I understand and can work through the maths to obtain the fact that the frequency is quantised and is inversely dependent on $L$, the length of the string, and $n$, some integer.
If I pluck a guitar string, this oscillates at the fundamental frequency, $n=1$. If I change to a different fret, I am changing $L$ and this is changing the frequency. Is it possible to get to higher modes ($n=2$, $n=3$ etc)? I don't understand how by plucking a string you could get to 1st or 2nd overtones. Are you just stuck in the $n=1$ mode? Or would the string needed to be oscillated (plucked) faster and faster to reach these modes?
 A: You always make harmonics when you pluck a string.  You can change the spectrum of the harmonics --- that is, their relative intensities --- by changing where on the guitar string you pluck.  
The usual place to strum a guitar is near the sound hole, about a third of the way from the bridge.  If you strum near the center of the string (12th/octave fret) you don't excite the first/octave harmonic, which has a node there, nor any harmonics in the half of the spectrum with a node at the center.
I think that strumming near the center of the strings makes the sound of the string become noticeably more "o"-shaped.  If you strum very close to the bridge, on the other hand, you excite lots of very high harmonics, and the sound becomes more "eee"-shaped.
You can excite only harmonics by lightly touching the string near one of their nodes.  Here's the method.  Pluck one of the strings like usual with the string open.  While the string is vibrating, slowly bring your finger down onto the string at the 12th/octave fret.  After you touch the string, but before you make contact between the string and the fret, you'll hear the pitch jump by an octave.  You've just killed the fundamental frequency, and all the harmonics with antinodes at the center, without much changing the harmonics with nodes at the center.  (Once you've figured out how much pressure to apply, there's no need to let the string sound open first; as another commenter points out this is called "flagolet".)
You can excite other harmonics as well at different frets.  The 7th fret (which divides the string into thirds) raises the tone by an octave plus a fifth; the 5th fret (which divides the string into quarters) raises the tone by two octaves.  My experience is that I can isolate higher harmonics on expensive guitars than I can on cheap guitars.
A: When you pluck the string you excite many many overtones, not just the fundamental.   You can observe this by  suppressing  the fundamental.  Pluck the string while holding a finger  lightly at the center of the string.  That point is an antinode for the fundamental and all odd harmonics, but a node for the even harmonics.  Putting your finger at that point damps the odd harmonics (especially the fundamental), but has little effect on the even harmonics. (There's a node at that point.)  You may have to experiment a little to find exactly the right spot and pressure.   Guitar players do this all the time to get a different sound out of the instrument.
