I usually start off with some handheld test shots in good light with the lens wide open. Will pick a subject at the near focus distance and also at infinity. This gives me a rough impression of whether I got a good sample or a bad sample.
Depending on the day of the week when the new gear arrives, I'll either take it for a spin if I'm going to be out and about with my family or do a more controlled evaluation on a tripod.
Similar to
@sean000, I usually shoot an aperture series of something flat with detail/texture/lines that fills the frame to assess for distortion and edge-to-edge sharpness within a week of getting a new lens. Both to see how the corners perform related to sharpness and also to see if there is any skew in the lens (i.e. is the performance uniform across the frame (even if the corners aren't as sharp as the center). Is either the right or left side of the frame not as sharp as the opposite side which can indicate skew (assuming you were shooting dead on to the subject)? Some lenses aren't sharp into the corners/edges (which is a lens design issue). If the right and left sides aren't symmetrical regarding sharpness then it's either a problem with your shooting setup or it's a problem with your camera and/or lens combo (i.e. skew--the lens isn't sitting perfectly perpendicular to the sensor).
One other thing to watch for relates to AF and DOF. Where is your camera/lens combo placing focus within the DOF when using AF? This ends up being a very complex topic that relates to AF on the camera body and also to how well the lens itself behaves regarding AF (could be either a lens design issue or a sample issue).
Say you have a 2 inch DOF for your aperture, focal length, and subject distance. Say you are shooting a person or animal where you want to get the eyes in perfect focus. The final pic won't be the same depending on where your AF places focus with a specific lens/body combo. It is possible that the AF will place focus on the "front end" of the DOF with the eyes being in focus and then extend 2 inches rearward leaving the nose out-of-focus. It's also possible that the AF will place focus on the "rear end" of the DOF with the eyes being in focus and then extend 2 inches frontwards getting the nose in focus but blurring the ears. In either case, the AF got it "right" by putting the eyes in focus, but the resulting images won't be the same. The assumption is that AF will place focus in the middle of the DOF or maybe 1/3 of the way in. This doesn't always happen however. Ideally you want *consistent* results.
This isn't just academic--my Sigma 40mm f/1.4 tends to focus on the front end of the DOF on my D850 which translates into effective back focus as the DOF using AF never extends as far forward as the math would lead one to expect. Worse, this behavior isn't consistent across various subject distances, so I had to come up with a compromise focus adjustment setting for the lens on my D850 that works "most" of the time. It does work as expected on my Z7.
Lens evaluation can be extremely complicated. It's easy to assume that when you get poor AF results that it's a "you" problem. Sometimes that is actually the case. But don't dismiss the possibility that it's actually a problem related to your particular lens sample. Always good to put any new lens through a controlled evaluation relatively soon after you get it so you are within the return window. Or testing a new body against lenses that you know are good. Cameras and lenses are expensive. Make sure they are working optimally when you get them. Nothing worse than getting poor results that are ultimately a gear sample problem and not really a "you" problem. But to determine that, you need to have a good understanding of what the expected behavior should be and then perform controlled experiments to confirm that your gear is performing as expected.
