Ways To Test iMac G3 Before Buying

I am going to buy an old iMac G3 Over the next couple of days/weeks, and I want to make sure that I am buying a system with no problems.

Assuming you're getting the Bondi-Blue iMac G3, you're looking at a machine with 20 years on it. Getting one with "no problems" is a tall order and akin to buying a "reliable classic car."

Some of the issues you can expect:

brittle plastic
malfunctioning CD-ROM
leaking capacitors
failing IDE/PATA Hard Drive

If you get a machine that boots and runs, you're well on your way. That said, this will run OS 8, 9 and OS X through 10.3 Panther.

What to look out for...

Make sure you have original install disks. If you do, the second disk will have your diagnostic tools. Boot that and run them
Run First Aid using Disk Utility. If the drive is actively failing, this will tell you.
Bring an audio CD with you to test out the drive and how well the speakers work
Determine whether you're getting the Rev A. or B model. The B version had the ability to support more memory (512M versus 384M) and better graphics (6MB VRAM on ATI Rage Pro versus 2M RAM on ATI Rage)

I do agree with @Allan largely.

But to answer/elaborate the specific question: what you'd ideally want to do is run AHT. These used to be included on a partition of the install disks a machine would ship with, but the AHT (Apple Hardware Test) disk images are still available online. And though the AHT support page index is gone, you can find it preserved at the Internet Archive. So I wouldn't consider the lack of original install disks "the end of the world".

I actually wouldn't necessarily consider a bad HD a deal-breaker myself, either, depending on the model. They still make IDE/PATA hard drives. And one should be able to find disk images of old OS X and even Mac OS 9 versions pretty easily. With a paid Developer account, I believe Apple still hosts a Jaguar image (which includes an OS 9 image). Moreover, there were a lot of revisions of "G3" iMacs and assuming it isn't the original Bondi Blue, you might find one that can FireWire 400 option-boot to an external drive.
However, I can't re-emphasize enough the point about brittle plastic, and, in general, the difficulty in teardown and reassembly with a machine like this. I used to be an ACMT at a AASP back in the day, and these guys were a pain to repair back when they were still covered by AppleCare, and that was with service manuals and genuine replacement parts. And:

Most importantly, it can be quite dangerous to disassemble. Unless you're quite comfortable with your knowledge of how to properly discharge a CRT safely, its yourself I would be worrying about first, before cracking one of these open. You would also need to be properly grounded for the rest of any surgery you attempt, because those boards will fry with even the slightest amount of static. And if you needed to Google "discharge a CRT"—don't even think about it.

On the other hand, if the CD drive is working and you've burned the right version of AHT onto a blank disc, and it runs, I'd say you're good to go. I don't know about you, but an original, first-gen iMac, with all discs, etc., and with everything in fully working order, would almost certainly be well above my price-point.

The main uses of the system should dictate a list of end-to-end tests. A couple examples:

Network/"Internet" access

I wouldn't get on the public network (none of your browsers will be secure), but you might want to at least have some transfer to your other home systems.

AirPort - it uses the older, original airport, it probably only talk to the old flying-saucer Airports.
Ethernet - your best bet. Everything is supposed to be 10mb compatible, but I'd bring the hub you will actually use, and at least get a green link-light on both sides.
Modem - This probably still works, but it depends on what you think of the words "hobby" vs. "fun".

Aging CD-drive

The HD drive concerns are valid, but also remember that your CD-drive is going to be getting old too. I had the CD-drive on a iMac G4 (lamp-style) fail, and soon after, the HD stopped spinning up, so for systems this old, you should think about both concerns.

Real mouse vs. "best" mouse

Some people loved the look of the "hocky-puck" mouse, others really disliked the feel. (I was neutral on that ), but I did find that a wired Mighty-mouse was my favorite mouse for this (it has "pill" shape that Apple stuck with when they upgraded).

If you will really use this system (for word processing of old games, etc.) you might want to get another mouse. You can even have two mice, if you need to compromise.

Printers

If you have legacy printer, of the right era, it will still work. With MacOS 10 you get a lot of language support out of the box, they make excellent non-internet "typewriters" for kids.