What I do is shoot some pictures!

Typically with a lens like the 150-600 my first step would be to concentrate on the performance at the long end of the range since that's where a zoom telephoto is usually weakest. I'd shoot wide open and stopped down (f6.3 and f8 in this case) and see if I could tell any difference between the shots. For a subject you want something with lots of fine detail, finer then the resolution limit of the lens. Birds feathers are good, animal fur, or failing available wildlife, distant road signs or bill boards with text small enough to be close to and past the resolution limit of the lens. I'd also shoot the same subjects from a closer distance (to equalize the image scale) with the lens zoomed back to 500mm and compare the results with 600mm. For a good lens the difference should be small.

I'd check center and edge, though for telephoto lenses I'd be mainly interested in center performance. I don't usually care to much about high sharpness at the edges and corners of a telephoto shot.

If you want to check out chromatic aberration, shoot something like powerlines against a cloudy sky, putting the lines parallel to and close to the edge of the frame.

You can check out vignetting by shooting a uniform target like the sky. This is easy with a telephoto, withe true wideangles you need a better target since sky brightness can vary over large angles.

I'd also check out the wide end of the lens for distortion, though that shouldn't be an issue with a telephoto zoom.

I'd check AF by comparing regular AF with live view AF and manual focus. Manual focus should be the best, followed by live view focus and then regular phase detection AF. Phase detection AF is fast, but statistically not as accurate as slower contrast detection or even slower MF using live view and 10x magnification. Phase detection AF may require AF microadjustment for best results. Contrast AF and manual are not affected by microfocus adjustment.

Obviously you should also check the other focal lengths, but most lenses perform pretty well in the middle of their range and are weakest at their shortest and longest settings. Typically distortion shows up at the short end (especially when the short end is wideangle) and softness or lowered contrast shows up the the telephoto end (usually due to uncorrected spherical aberration).

I don't know if you shoot FF or crop sensor, but the vast majority of FF lenses perform very well on a crop sensor camera since all aberrations except for spherical aberration get worse as you move out from the center of the frame. If you really want to see how well a lens performs, look at the corners of the image taken with a full frame sensor. However from a practical viewpoint, unless you'res shooting flat artwork or wideangle landscapes that need corner to corner sharpness, the performance in the center is much more important than the performance in the corners.

If fast AF tracking for sports and birds in flight is important, your best test is on the actual subjects you want to shoot.

I'd also shoot the same subject with IS on and off at several focal lengths and shutter speeds to get some idea on how effective the IS system is. 2 stops is reasonable, 3 stops is good, 4 stops is excellent (and rare). Manufacturers all seem to say "up to 4 stops" these days, but 4 stops is rarely the case in practice.

Thanks Bob.

I shoot crop with a T3i.

Focus accuracy is kind of important since I have no micro-adjustment available. With the 150-500 I had some sharpness issues at an airshow, which, of course can be a lot of things.

Thanks, Bob, Maybe I will try to catch some airplanes.

Does the tamron have panning modes or auto-pan sensing?

Thanks, Bob.
I got the lens and my quick evaluation looked pretty good so I kept it. (Note to self: Turn off AUTO ISO before performing test, doh!)

Incidentally, both Rockwell and Luminous Landscape liked this lens, one of the the few times those sites agreed on anything!