If you're a good DJ or stylist, with an immediate sense of whether something works or doesn't, then you're halfway towards being a good font picker. Being able to pick out what's right among a sea of options is a skill that informs any kind of art and design, and many pieces of daily life. If you can recognize yourself doing it in one way, it's easy to apply it to any selection process.
The considerations are the easy part. I usually know whether I'm looking for a display or a serif font. I know what size it's going to be. I have some idea what kind of other fonts it's going to be paired with.
Considerations are good for quick elimination, narrowing down the search.
There's another aspect to it though, and it has to do with style, feel, energy, vibes, whatever you want to call it. It has to do with taste, but when picking fonts, I have to set my taste aside, while also trying not to anticipate the taste of my audience. It's almost like you have to assign a personality to the content, and that personality has its own taste, and that's what needs to be tapped into.
When I'm looking for a title font, I'll put the company name, band name, page title, whatever it is, into whatever sample text thing I can find, which in my case is usually the one in Google's font directory, and buzz through samples. I do it really fast. It used to take forever. I'd stop on way too many options and ponder them. I didn't know how to say yay or nay or maybe. The biggest problem with that is it takes way too long and the results aren't that great. So I had to learn to spot the potentials like red balloons in the sky, and add them to my list.
Going through my list used to take forever, too. It was always way too big. Some projects are easier to font-style than others, but these days my lists are pretty short, and it's down to a shortlist with some quick eliminations. Once I see the first round of selections all together, it's obvious which ones are no good.
If I'm making a font selection for somebody else, I don't show them anything before my shortlist. And I have to meditate on that shortlist a bit. I may have to do a lot of work with any of those fonts, so it's important that I can completely endorse all of them.
Once I get my title font, I pick the body text font. I set the display font up so I can see it while I'm scrolling through the massive list of possible serif fonts. If I can, I use serif fonts for paragraph text because they read faster in a paragraph, and the opposite for headlines. Some web designers think that serif text doesn't look good on screens, and for a time that was true, but that time is over. I explain that in this other article about web type.
Then there's always some little bitty fonts to pick. When I can, I use a "humanist" typeface like the classic Verdana or some fancier, newer font stack like the stuff I talk about in my article about ultra neutral fonts. This is the stuff that's actually going to be as small as possible, so it's not going to work as anything but a sans-serif font, and a plain, round one at that. Some fonts are designed to look great big, and some are designed to look great small. Incredibly, the most hated font ever, Comic Sans, looks decent and is very legible at very small sizes. It's one of those fonts that looks worse at every point size above about 8.
So there's usually 4 sizes of fonts to a document: Your main title, headings, paragraphs, and small print. You may need a few different sizes of headings, and they need to be noticeably different, but not get smaller than the paragraph text, or larger than the title text. And if anything, it should stay a safe distance from both.
The amount of variation in size between different text elements really depends on what kind of document you're styling. It's a stylistic consideration, it's a technical consideration, but it's also easy to tell if the variation is too much or too little. It has to be tackled, though. If the size relationships of different type elements isn't right, it's one of those things that makes a document look cheap but you can't tell why. Some people apply the golden ratio. Keep in mind, font size is deceptive, so you actually have to measure the text.
I start the design phase of most web projects by picking fonts. It's a good place to start. It's a good way to initiate the client into the working relationship, and establish some precedents.
