This actually indicates that you had too little resistance on the rear tire. Without enough resistance from the fly wheel, your tire will slip, get hot, and degrade. All trainer brands are a bit different, but the rule of thumb I use is to make contact between the tire and fly wheel, then turn the knob for the fly wheel 4 more times. You want to see an indent in the tire that's about 1/4" deep.
Other tips -- clean off your tire before using it on the trainer. I use windex. You can also use rubbing alcohol or water. Always inflate your rear tire to your regular pressure (probably 100-110psi on a road tire). An underinflated tire will also wear out quicker.
As far as tightening the clamps that hold the bike in place, you want it tight enough so the bike won't fall off the trainer but not so tight that you can't un-clamp the clamps.
Hope this helps. Oh, and if you know the brand and model of your trainer, you can find the users manual on-line at the brand site. Just google it.