Reboot and select proper boot device

Hi! I'm new here, and I fear this area of computer technology is something I'm too ignorant of, so bear with me. But our laptop of late has frequently taken to randomly shutting down and then when you try to reboot it, the screen has a message that includes something about reboot and selecting the proper boot device or inserting a boot device. By holding the power button and/or selecting ctrl/alt/del from that screen (the ctrl/alt/del method seems to work when the power button doesn't), we have repeatedly successfully gotten the computer to reboot properly and the computer works (albeit, it seems to be really slow these days) until the next time it randomly happens (and it seems to be happening more often than it did at first). Sometimes it happens when the laptop is plugged in, though often it's when the laptop is unplugged, or sometimes it happens immediately after we would unplug it. It does seem the battery life of the laptop isn't as good as when the laptop was new.

Where to start with diagnosing cause and then of course, method of repair? Thanks!

Since when do you have the laptop
Have you replaced the battery with a new one since you have the laptop?

Normally a battery is dying within a few years, 2 to 4 years (4 with extreme care).

Please read Battery Usage - Generate report.
This tutorial gives you information about how your battery is doing with its capacity.

We've had the laptop 1 and 1/2 years, though as I recall it had been refurbished, so maybe the battery is older than that. Am I understanding you to say there could be a connection between the waning battery and the boot device messages? I ran the report......would screen shots of portions of it be helpful for you?

This part is most important of the tutorial:

Sorry if I'm being "slow".....here is the screen shot of mine. I'm ignorant enough that I don't know what to do with this info., though.

Your battery is at about 33% of what it used to be.
It would be smart to replace the battery and see if that helps.