Right-click on the track and select the track height.
You, of course, also have the option of zooming in using the + and - keys.
To sum up, in timeline view:
Method (1) RIGHT-CLICK track and then menu-select: small, medium, or large.
Method (2) There is also a vertical re-size handle, but it only appears in the grey area 'Video 1, Audio 1' (etc), on FAR LEFT. NOT on the track itself. Unfortunately with this grey skin there's no obvious handle to grab. Just have to hover around until you find it. Then LEFT CLICK, hold down, drag.
Method (3): Your other suggestion of Zoom +- did not work in my version of PE8.
+ and - will zoom in and out if you have the timeline panel selected.
But if it doesn't you can always use the zoom slider along the top of the timeline.
(BTW, per my book "The Muvipix.com Guide to Premiere Elements 8", you should also go to the Window and select Show Docking Headers. This will display a gray bar along the top of each panel. On the right end of each Docking Header is a >> pop-out menu. Click on it and you'll get easy access to some additional tools for each panel.)
Method 2 is the way that I vertically resize a Track, but it can be a bit of a "pixel-hunt," as you need to slowly move the cursor in the Track Header (that area you identified to the left of the Track with the name, Display and Keyframe buttons). After you get this a couple of times, you'll soon pick up on exactly where it is (yes, the GUI does not make this easy), and can nail it within a pixel, or two. When you get the resize icon, that double-line with up/down arrowheads, just click-drag to suit.
With PrPro, one can also use methods, like tilt-roll on mouse wheels, etc., but I have not found a way to program/set Preference, with PrE. However, I do not have PrE 8, so do not know all possibilities in it. [Haven't gotten that far into Steve's PrE 8 book yet!]