What are the ways people can/do people invest in art?
You invest in art by purchasing art. Christie's and Sotheby's are the two "world-famous" auctioneers of antiques, "old" art (for want of a term) and modern art, but there are many regional auction houses and specialists in almost everything (whether "vintage cars", "books", etc etc etc). There's also (for want of a better term) "novelty" art, things like baseball cards, cels from Animes, etc etc (some of which have performed incredibly well indeed over the last century)
What strategies are there?
The strategy to all investing is that
you guess what you think will go up. (Example: real estate: you think Seattle real estate is going up and LA real estate is going down.)
you then buy that thing. (In the example, you'd buy Seattle real estate.)
This is the only "strategy" to investing. You are guessing what will go up.
This applies identically whether you are investing in real estate, antiques, art, stocks, commodities, gems, wine, or any other category.
Yes, there are many "art" funds in all sorts of categories (antiques, gems, modern art, classical art, etc etc).
Don't forget that stocks (ie, the world's couple thousand biggest stocks) are very liquid, and things like physical gold are very liquid; whereas real estate and "art" is less liquid. It can take ages to sell some land or say paintings.
(Similarly things like stocks/bullion have very low transaction costs, whereas physical tangibles (houses, Picassos etc) usually have high transaction costs.)
Conversely, "art" (antiques, paintings, etc etc) are tangible (they are absolutely real - precisely N Vermeers exist, and that's it) whereas stock instruments are just a thin-air abstraction.
as you imply there is really no "day trading" of tangibles to speak of. One example is the market in wine (whether "futures" or cases); which was pretty hot for awhile and you could trade it in a rapid, day-trading-esque manner. But yes there is really no fast, day-trading analogue for tangibles like antiques or real estate. (That being said there are REITs which you can trade pretty fast, but they fall apart in banking/real estate crisis.)
The raison d'etre of tangibles is that you hold them in your hand (well, safe!) and that's the end of it - no "crisis", broker, regulation etc can take them away. This is probably why historically all "wealthy families" in all socio-historic settings, have held a large amount of their wealth in physical tangibles (ie - art, antiques, etc, as well as of course real estate, with the caveat that you only "own" real estate due to a sovereign power saying you do).