This site about modern Greek has a guide for handwriting the Greek letters apparently based on how people actually write in modern Greece and Cyprus.
It can sometimes be useful to think about how the lowercase letter might have developed from the uppercase letter through people attempting to write it faster, smaller and without lifting the pen off the page.
The uppercase zeta is just a Z and when writing it smaller and faster you can imagine emphasising the angle at the top and curving the angle at the bottom, then adding a little kick so it doesn't look like a 7. The uppercase xi is three horizontal lines and the lowercase xi is an attempt to draw these three horizontal lines without lifting your pen, but in the end is just a zeta with an extra kink in the middle.
Lowercase sigma is a bit tricky to make it not look like a 6 and I still haven't perfected that yet. The webpage above draws the circle anticlockwise and then changes direction to put the top line. I see other people drawing something like a capital u starting from the right hand side and then making a sharp bend (but not a kink) to draw the flat top. I like to start at the far right and draw the flat top right-to-left then do the loop, imagining that you started to write a capital sigma from the top but then gave up after the first bend.
Concerning epsilon, I don't like to use $\epsilon$ because it looks too much like the "in" symbol $\in$. It also really really bugs me when people use the curly epsilon instead of an in symbol.
Finally I have noticed the use of the two phis, but never both by the same lecturer. I would not recommend indiscriminately switching between them but sticking to one so as not to confuse whoever was watching.