Should I buy a new mac mini, or buy a refurbished desktop/laptop?

I don't know exactly which model Mini you are interested in buying. Since you mention it has been upgraded, that is only possible with Minis dating to 2012 or earlier. Realize these are now 6 years old or so and I don't know for how long Apple will be issuing OS versions that support computers that age. Now I have almost never run a current MacOS version so it wouldn't make any difference to me, but if you do try to stay current then this may narrow down the selections for you. Also keep in mind that newer Minis are more or less locked into their configuration, as are all Apple computers (with a few possibilities to upgrade RAM, but only in a few models). Personally I find a 2012 Mini more desirable than later models but you are soon reaching a time when these will be showing their age and Apple will stop supporting them. Everymac.com is a good web site for getting detailed specifications on all Mac models, including what can actually be done with them now versus Apple's specs. pages which tend to reflect only information at time of release.

FWIW. Although I have a MacBook Pro, I use a Mac mini as my "main" computer. I currently have three of them for various uses but use my Late 2014 model for most of my computing needs. Everything you listed can easily be done with the mini.

Regardless, be sure to get your mini from a reputable source and know their return policies in case you have issues with it. If you decide to go with a refurbished unit, I suggest that you go with either Apple or Other World Computing for your source. If you are the type that likes to do it yourself upgrades, and given the choice between a 2012 & 2014 model, go with the earlier one.

Tim Cook indicated last Winter that Apple was not yet done with the Mac mini. Like the rest of us users, I don't know what Apple's product roadmap holds for a release of a new Mac mini. If I were in the market for that form factor, I would hold off until Apple has addressed any Fall Desktop updates (if any).

Why pay full price for a four-year old Mac mini with significantly outdated components. Will it just get dropped off the supported list for macOS 10.15?

Unfortunately SadCanary never really responded to my question and told us which specific model Mini was being considered, just general specifications with a lot of mention of "upgrading". As far as I know the 2014 Minis are not upgrade-able and I agree with you I would not buy one, particularly if it was in base configuration with 4 GB RAM. They are also a 4 to 5 year old model and soon to be added to Apple's forgotten-ware list. I also don't think they have particularly impressive specifications and if you're going to get something in the vintage list you might as well go for a late 2012 model which can be upgraded nicely. Yes, you may be missing a few tweaks the 2014s have but you're getting a machine which can be upgraded and repaired. The only downside is they are now in demand because of this and not particularly cheap for a 6 year old computer.

I have one of the first Fusion drives (a 2012 iMac) and don't regret the purchase. Today I wouldn't consider buying a Fusion drive simply because the problem it solved no longer exists - SSD prices have plummeted. Even if I could only afford the 256GB SSD, that's what I'd buy and if I needed additional storage I'd get an external drive for multi-media. Pictures and music don't need a fast drive the operating system and applications do.

The mini is a great little computer but like many others here I wouldn't purchase a new one today - they've gotten no attention in nearly 4 years and can't be upgraded. If I needed an inexpensive reliable Mac and couldn't wait for the "someday" mini resurrection, I'd look at an iMac refurb. Don't get me wrong, I love my MacBook Pro but you get more bang for your buck in a desktop than you do a notebook.

Yes. the 2012s are nice machines—we have two of them here, one of which I bought right after the 2014s were announced, to a fair amount of disappointment among the Mac faithful. Used ones still have very good resale value.

Seems like there must be a cheaper and easier way to just try out working with the Mac OS. I would not buy a 2014 Mini.