Tuning snare drums took me the longest to learn. Follow the link to Prof Sound's Drum Tuning Bible and read. It takes a while to understand the tuning zones he writes about, but the more you play with the tuning of the batter head and the snare side head the more you will understand. Some of these zones are very fine increases or decreases in tuning, so take your time. Some of my snares have the batter tuned higher than the snare side, while others have the sound I want with the batter tuned lower than the snare side. Don't be afraid to experiement, but do keep notes to help you find certain sounds again.
Too much ring (overtones) can be damped by using drum rings, moon gel, or my favorite: tuning. You will find that as you go through tuning zones you will get major ring, moderate ring, or very little ring. Once you get close to the amount of ring you want, you can detune (slightly)one lug on the batter and that will kill any remaining ring.
My approach to tuning: Seat the heads (see Prof Sound), tune the batter for the tension/rebound/action you want, make sure the lugs are in tune with each other, then tune the snare side head through the different zones until you get the sound you want, again make sure the lugs are in tune with each other, then adjust the snare wire tension and alignment
click here , finally adjust the overtones a hair with detuning a single lug. I hope this helps.