You do want a coarse grind for french press coffee. The key is letting the coffee steep for long enough to make a strong brew. I have good results with a 5 minute steep time. Be sure to use a timer, and compare different times to find the optimal brew. As a general rule, if you are at altitude, then you need to add time because boiling water is cooler. (Note that I am at 5000ft, so 5 minutes may be too long for you.)
Below is a picture of the coffee chunks in a coarse grind on top of a US 5 cent coin (21.21 mm in diameter).
The picture is from this article.
EDIT: The reason to use a coarse grind for a french press is that finer grinds will get stuck in the filter and/or push their way through. This has three potential problems: 1) it will be harder to press down the filter, possibly leading to spillage or breakage; 2) your coffee will have grinds in it; 3) the coffee may become bitter, as the finer grinds that passed through the filter remain in contact with the coffee for too long.
There are a lot of good detailed answers on this page already, but I guess I could add my own $0.02 to the conversation and maybe a different angle to the answer..
I have been home roasting and brewing for about 2 years now and I think the #1 thing I've learned about pulling espresso shots is that it is all about consistency.
In order to effect consistency you have to control your variables, and the more variables you get control of, the better chance you have of maintaining consistency.
Over the past two years I have continuously refined my process by modifying the entire chain of coffee production, whether it meant ordering a new piece of hardware or changing how I did something.
For instance, I installed a PID on my Racilio Silvia in order to control brew temperature, I bought a naked portafilter to get better feedback on the quality of my shots, and bought a pressure gauge to make sure I was getting proper brew pressure at the group head, the list goes on and on.
So when I saw your question about "always using the same coffee" I interpreted it as just another one of those variables that you can get control of and that will effect the quality of the shots you pull.
In the beginning I stuck with the same coffee and roast to try to minimize the number of variables I was working with. Once I was more familiar with how each variable effected my shot and how I could manipulate them to get what I wanted, I began to experiment.
So I guess for me keeping the coffee the same boils down to removing some noise from the system to make it easier to manage...
Best Answer
Brewing espresso is particularly susceptible to problems relating to an inconsistent grind. I am not familiar with a burr mill specifically, however burr mills are generally recommended for espresso grinds as they tend to provide consistency. Bear in mind that the grind can either be too coarse and the water will run through without much extraction due to the pressure and space between the grinds, or it can be too fine and allow for a backup of water and espresso water and grinds all over your counter.
NOTE: The pressure with which you pack/tamp the puck (the grinds when pressed in the brewing basket form a puck) will also play an important role in the extraction process and the crema formation. You can get a feel for how hard to tamp the puck by getting a scale and pressing down until you hit about 30 pounds of pressure; you want to pack the puck at that pressure of tamp.