A variant of the famous ideal gas law (PV=nRT) is the relationship that pressure is proportional to density times temperature. That is, more or less, P = D * T.
You're right, as temperature goes up, density drops (volume increases), AT A FIXED PRESSURE.
As we go down in the atmosphere, the air does indeed get hotter (T goes up), which would make you think that density should go down (D goes down), which is the same thing as volume increasing.
BUT, as commented by another person, pressure goes up as we go down in the atmosphere as well. If P goes up, then D * T has to go up by a proportionate amount. Analysis of actual atmospheric conditions shows that for every 100 "units" of increase in P, D goes approximately up by 80 units, and T goes approximately up by 20 units. That is, Density goes up (volume goes down) by about four times as much as Temperature rises, per unit increase in Pressure. It's a good thing that density takes so much of the "brunt" of the increase in pressure... otherwise it would be a lot hotter at the Earth's surface, like well over boiling!
This too all confused me until I started classifying in my mind that volume and density are essentially the same thing (since volume is inverse to density, and vice versa), and that pressure is conceptually different than density. They're related by temperature, but that's the only relation. Just remember, P is proportional to D times T. Just a few more conceptual examples: If D goes up and T goes down by the same amount, P remains unchanged. If P goes up then either D or T has to increase. If P goes up and, for example, D decreases, than T has to increase by even more to "equal out" the increase in P.
Hope this helps! The figures for the "standard atmosphere" (pressure, density, and temperature for a given altitude) that you can use to play around with to help you understand it better conceptually (at least it helped me out anyway,) can be found at http://www.usatoday.com/weather/wstdatmo.htm
. I'm happy to try to answer any further questions, or to elaborate further on this question, if anyone's interested. I'm trying to learn weather very well myself as I'm thinking about being a dispatcher someday in the future, so the better I can teach it, the better I understand it thoroughly myself!