Well, what kind of sound are you trying to get? Normally for a side-snare sound, there will be a single-ply coated head (for more attack) tuned medium-high to high. Try tightening the snare-side head to get more snare response (and less of a mushy sound).
Straps vs cables for snares - there's not that much of a difference. You'll get a little bit better contact with cables, but they'll chew up your snare beds. Also, don't overtighten your snares. It'll gate the snare-side head and can actually make the drum quieter. You might even want to try getting a new snare-side head - I recommend the Remo Hazy Ambassador and Aquarian Classic Clear (be sure they're both snare-side heads, which are 2-to-3-mil single-ply heads)
Also, make sure both heads are tuned evenly. Tap around the head about an inch inward from each tension rod, and tune each spot to the same pitch. That helps more than most people think, since it will rid the drum of bad overtones almost completely.
Hope that helps!