How much is contents (or finding out what that is) worth to you? And how much time/effort you want to spend on it?
The 'treasure trove' aspect of such a find is usually overrated: it sat in storage for a long time without negative effect. If contents can be retrieved (likely!) it will take time to sort through.
In terms of valuable:
1) Personal stuff. But remember it's daaated. For that reason alone there could be 0 value in recovery.
2) Unique software: titles that somehow have not found their way into online archives. For any reasonably popular machine like TRS-80, this is more rare than one may think. But if you were a programmer back in the day, source code finds may be very welcomed by the community.
If disks are labeled properly, you could do a quick search through online archives, to decide whether a title is rare/special enough to spend time on recovery. Already a good copy out there? No need to spend any time on it. Titles may not be online, but with good copies existing 'offline'. A quick email / forum post may get you a copy easier than trying to read >30y old floppy.
Recovery: for each floppy, save both files and a full disk image. At least until you've figured out what contents is/does. Software that looks like regular files, may access a disk in other ways than through a filesystem. Some software may come with copy-protection schemes - good luck with that! ;-)
Once you have disk images, there's likely 1001 tools out there to process those. But establish a workflow with a few disks first: find out 1st hand what equipment / process / software tools work, then proceed with the rest. As opposed to: read 100s of floppies, only to discover afterwards that tool used read only 1 side of 2-sided disks. Or things like that. You may perform multiple reads of a disk, and (binary) compare results. Or give retrieved software a test-run on emulator(s) to check for obvious faults.
> Is there a risk of only getting one shot at reading them, as with some magnetic tape?
In my experience: no. Floppies are fairly robust as far as magnetic storage goes.
Equipment: the older, the harder it'll be to find a working setup. Unless it's quick/cheap/easy (like obtaining a USB 3.5" floppy drive), don't bother. Find person / organisation (computer museum?) that have a working setup & know how to use it. If money is no object: data recovery company can do it for you ($$).
If you are confident about disk formats, that can be of great help. Eg. I've read of some USB floppy drives that only read HD floppies, but don't deal with DD floppies. Similar considerations for software tools / OS support etc.