I`m a newbie to this board and to detailing in general....
Welcome to Autopia!
.. We have a Toyota SUV that`s been the grocery grabber for years. Although it is old, the driveline and interior are still in great shape. Only the original factory white and dent free exterior looks shabby because of the oxidized white paint, some black overspray it picked up at the high school auto shop and dozens of paint chips on the nose and hood, some of which started to rust.
I have more time than $$$ right now and decided it would be a good learning experience to try bringing the exterior back to life myself...
Ah, that sounds like a great detailing project, and actually quite similar to what I had to do when I got my beater-Tahoe.
Through a guy at work, I have access to a Flex orbital and DA buffer and I have a small budget for detailing tools, polishes and waxes.
Please provide more info about which Flex you have access to. Maybe you do/don`t need to buy your own.
Compare my responses to the others, some of the areas of dis/agreement might be interesting.
Should someone like me with zero experience attempt to use these buffers or am I likely to do more damage than good?
IF you approach this with the right mindset and a decent knowledge-base you should do fine. I know plenty of people with zero experience who have, but the operative word there is "IF".
Is it possible to make a difference in this old paint if I stick to elbow grease?
Possible yes, but not in the practical sense. How many dozens of hours...and that`s sure no exaggeration, probably understating it...are you planning to put into that "elbow grease"? The polisher is a practical necessity if you only have a few dozen hours to spend on that aspect of this (again, that`s no exaggeration, or at least not much of one...people vastly underestimate how long this stuff can take).
What is a good way to fill in the chips in the paint and get rid of rust in the chips?
If at all possible just pay to have the hood repainted. If it`s *really* bad then that`s simply what you need to do even if it means waiting a long time while you save up the money (Autopian Xtremekustomz could tell you how to DIY a repaint if you`re interested in trying that, maybe worth considering if it`s really bad). IF repainting is not possible, then I would:
-Grind the rust out of the chips using a rotary tool ("Dremel") and tiny diamond burrs. Yeah, there`s that "need for the right mindset and knowledge-base" again

-Prime the ground chips with a rust-killer product (but something better than POR15) giving adequate consideration to topcoat adhesion. FWIW, I`d probably recommend Rust Bullet. I would not recommend a conventional primer even though you`ve ground out the rust
-Touch up with DrColorChips, but use it like a conventional touchup paint instead of following their "smear it all over" instructions. Yeah, that can take a while.
-Level the touchups with the DrColorChips solvent
But see alternative below.
Should I take care of the chips first and then polish or the other way around?
Do the chips first so the rust can`t dislodge while polishing and get stuck to your pad/towel and thus cause new marring.
ALTERNATIVE- WARNING, Autopian Heresy follows! OR...do a thorough chemical decontamination. Hand-compound the rusty chips until they`re not so bad and *any* loose rust is removed. Then just do the rest of the compounding/polishing (note that is gonna be a two-step process no matter what) giving consideration to product residue getting trapped in the chips. Then apply your Last Step Product ("LSP", i.e., wax/etc.), again giving consideration to product residue getting stuck in the chips. Yeah...this will leave slightly surface-rusted paintchips. But that`s not necessarily a problem; I haven`t touched up the ones on my Tahoe and they`re still OK after years of use (including winter); I left some chips like that on my jaguar for *decades* without any problems (my painter finally touched them up because he couldn`t stand seeing them any more).
How can I deal with the black overspray? It has been on the paint for years and I can`t even get it off with my finger nail.
See if the Chemical Decontamination helps any. Otherwise see about Overspray Clay. And the compounding should help too.
If I can use a buffer, what pads/polishes/waxes should I buy? I don`t mind spending more for quality as I only have one vehicle to do.
I`ll let others handle the "which pads?" as I`m out-of-date there. For the Compound I`d use/recommend Meguiar`s M101 and for the Polish HD Polish. OR the Griot`s equivalents (IIRC they`re called Fast Correction Cream and Finishing [something]).
If I do end up polishing the 20 year old paint, will I have removed so much of finish that the paint won`t last long after that?
Well, you won`t be doing it any favors

But just take it easy and settle for "significantly better" rather than aiming for perfection (and that`s likely how it`ll turn out anyhow after you`ve spent a few of those hours on the first panel).
I don`t know whether your vehicle is one of the rare ones with Single Stage paint or whether it has Basecoat/Clearcoat paint. Find out before you start. "How do I do that?"...heh heh, there`s a good way to test your knowledge-base and mindset

Sorry, couldn`t resist....