How do I fix a shower mixer so I get hot water without turning the knob all the way?
This depends on the shower valve you have.  

On some there is nothing you can do short of replacing it.  On others there are pressure balancing settings that you can change and these should be adjustable once you take off of the trim.  Most of the time they are right in front and you will see notches. 

For American Standard valve they call it the "hot limit stop".  The picture I have below shows how to change it.  You will need some pliers (to turn on the shower without the trim on it) and a screw driver (to take off trim and adjust setting).   

This is kind of hit and miss with most valves - you really just have to change it and see if it is what you want.  I have mine on the hottest setting and the range is kind of warm to pretty hot.  Can't take a really cold shower in one of my bathrooms!

Warning:  If you have children using the shower then I would really test the settings for scalding water.  Most newer ones that I have installed don't allow all hot to got through but who knows.  Test it out.

Also if you look like you could or might break something when you are adjusting the valve, I would leave it alone until you can figure out what your replacement options are.  You might want to figure out what model it is so you can look up instructions and then to have that model # to replace if needed.  Changing this is a very easy process for new valves but the much older ones can have issues.