Audio in Premiere Pro is different--but quite a lot, in many circumstances. I'd recommend going through the help file's audio section to get a better idea of what some of the main controls are: Adobe Premiere Pro CS5 & CS5.5 * Editing Audio
But to answer your question more directly: the Audio Mixer controls track volume, not clip volume. By default, the Audio Mixer isn't set to drop keyframes, so it will affect the level of a track as a whole; check out Recording audio mixes to get more background on setting up the Audio Mixer for automation, so that you can record level changes.
The rubberbands (the yellow lines you're referring to) can be more finitely controlled by holding down the Ctrl/Cmd key as you drag; the values will move in smaller increments, then.
You don't have to load a clip into the Source Monitor to adjust the levels, if you want to adjust by the numbers. Just single click a clip in the sequence, and if the Effects Controls panel is forward, you'll be able to adjust the volume with the standard effects sliders/dials. Note that keyframing is enabled by default for the Level property, so if all you want to do is adjust the total level of a clip, it might be faster to do it in the sequence.
Audio mixing in Premiere Pro is pretty capable and the feature set is pretty deep, but I'll grant you it's a little overwhelming coming from a different NLE. I don't know if any of the previous helps you, but post back with any specific questions and we'll try to figure out a way to approach this in a fashion that makes sense to you.
You might want to also see this ARTICLE on adjusting Audio Levels in PrPro. It gives a bit of background, but I highly recommend spending some time with the Help Files, as they go into far more detail, and with the Audio Mixer, that detail can be very, very useful and important.
Thanks Colin & Bill.
I still think Avid & FCP have the right system for audio mixing but those tips you gave me will help me adjust.