I'd think just using the phone as you normally would for a week would be enough to figure out if there are any hardware issues with it. Also, since the phone is a new model, if there are any issues that can be solved with a software/firmware update it might take some more time for Google to release those updates.
I've read some people have had connectivity problems. Not sure if the people who had those problems have a hardware or software problem. But if I were going to try and test for hardware problems I'd do the following
- test connectivity using both wifi and mobile data (could do this while testing others below)
- stream video for a few hours
- make and receive phone calls and talk for a while
- video calls
- use bluetooth headphones
- play a game that requires a lot of processing power to run
- test wireless charging if you can or care about that
- take photos/videos utilizing each available camera
- transfer large files to and from a usb drive to see if the usb-c port has any problems
- (after the previous one) install GrapheneOS
Those are just things that I could think of off the top of my head. Maybe others have more suggestions.