This is highly individual. I never go somewhere to "learn". (I completely agree that you can read the Wikipedia, and most of the time I don't even care.) Some people seem to have different preferences, hence the profession of a tour guide.
From the perspective of feelings, being somewhere provides you the full 3D experience, which is stronger than just seeing a picture on the screen. Even watching things in a museum or a gallery feels different than looking at pictures on the screen.
Some places provide you things you don't have at home: a sea, a mountain, a forest, a jungle... My typical vacation is going somewhere close to nature.
Sometimes you want to experience a different culture. Could be any kind of difference, just to have a novel experience; or could be a specific culture that you enjoy. People have emotional associations with some cultures, based on stereotypes, books, and movies: romance, adventure... You may want to practice a language.
Traveling somewhere means that you don't sleep at home. If you stay at a hotel, you have to pay for the bed and food, but you don't have to do the dishes and other annoying work; it is expensive, but pleasant. (Yeah, you could also rent a hotel room right next to your house, but this way you combine the necessity of living somewhere on a vacation with the luxury of having more service.)
And then there are the thousand different details, many of them unpredictable, that you can find in a different place. You may find a nice park, a nice cafeteria, or perhaps just a nice view, at the new place. Around your house, you probably already know most of these things.
Sometimes you go to a specific place to meet people who live there. Sometimes you coordinate with people who live far from you, to spend the vacation together at the same place.
...I probably forgot a few things here, but the idea is that a vacation is many kinds of things. Different people put different weight on individual components, thus you get different kinds of vacations (hotel vs camp, distant country vs nearby place, etc.).
Does visiting family count as a holiday in the relevant sense?
Depends. It will be closer to the "archetypal vacation" if they live far away from you, in a different kind of place (e.g. big town vs village), if their neighborhood is interesting, etc. But there is no official definition, and in real life it probably depends on how your partner feels about it.
How much money should I be willing to spend on holidays?
Depends on your income and spending habits. And what type of experience you aim for. Foreign travel will be expensive; local train can be cheap. Hotel will be expensive; you can also find a cheap accommodation. Some countries or regions are expensive, some are cheap.
My algorithm is something like: First, choose the type of experience, e.g. "somewhere near a forest, where I can take walks in multiple directions; also not one of the places I was at recently". Then choose the minimum level of comfort you can live with, e.g. "no cooking, a comfortable bed and a warm shower". Then look for the options, prioritizing for seeming interesting and being cheaper.
Is 'holiday' a coherent enough category that I can treat it as primitive for the purpose of this question?
Yeah, I think so. "Going to a different place, with the intention to have pleasant and/or novel experience." But of course some examples are more central (a hotel at the beach; a hike in the mountains; a guided tour in a foreign country).
Seems to me that "not being at home" itself makes a large part of the definition, although people can also go on business trips. (And you can make an edge case, such as a business trip in an exotic country, where the work is mostly a pretext to be there.)
feel free to answer ones not listed here
Try different things, find out what makes you happy. If you want to go with other people, different kinds of vacation may require different people.