Since this is a how-to question, I will answer it with practical advice. Questions about the particular halachos mentioned in passing may be asked separately.
Here's what I've been told to do, with illustrative pictures. Before changing anything about what you do personally, you should talk to both your rabbi and your doctor. What is written here is just practical advice, gleaned from years of experience (and conversations with Rabbi Hirsch Meisels).
What are we dealing with?
A: blood sugar meter (measures blood sugar levels)
B1: strip container (contains strips)
B2: testing strips (brings blood into meter)
C: lancet (draws blood)
In order to use the meter, the strip has to be inserted. Doing so turns the meter on, and possibly created a kli (vessel), as neither piece is usable by itself. As such, a shinui should be used to insert the strip into the meter.
Meter with strip in it:
Suggested shinui (insert halfway, and push against a table):
Drawing blood ("lancing")
Drawing blood to test is usually done by pressing the button on the lancet with a finger, like this:
To replace this possible איסור דאורייתא (Biblical prohibition) with a שינוי (different way of doing it, considered halachically to be less severe), lancing should be done with a knuckle, like this:
Anecdotally, I've used about everything, from elbows to forehead. (Yes, forehead :P )
Squeezing blood out of the wound (in case there is not enough from just the lancing) is problematic on Shabbos, due to the prohibition of dash (the weirdest form of squeezing liquid from solid you're ever likely to see), so it should be done with a shinui if possible.
Suggested shinui: use the lancet
Then put the drop of blood on the strip, like normal, and collect the result. Do not turn off the meter after use, because this serves no purpose on Shabbos (unless the battery will die, and this meter is needed for more tests over the course of Shabbos).