This is a great question and I have been enjoying thinking about it.
My art purchases are actually limited not so much by budget as by display space. Between the small child and my preference for visually uncluttered views, I just don't have much room for art beyond what we already have. So the first thing I would spend the budget on would be appropriately climate-controlled storage, perhaps some additional display space, and a part-time assistant to come rotate things out every month or so.
If I have additional display space I am effectively running a very small museum already, so I may as well open it up to the public. Hi, public! I will set my imaginary assistant to cataloging the collection and getting high-quality photos online, as time is available. A museum needs a more focused goal than The Museum of Stuff I Like, so I think I will focus on local contemporary artists, meaning I will go around to the various Art and Foodstuff Festivals and browse amongst them, as I am wont to do, and then I will actually buy stuff and bring it home with me, which I am not so much wont to do at this time. So it is maybe a subset of The Museum of Stuff I Like, but I think the focus will improve the collective experience.
My particular tastes in Art tend towards natural materials, organic forms, and things that hover in the fuzzy area between Art and Craft. We the museum will have really nice local-artist chairs and benches, upon which you the public will be encouraged to sit. We will have a lot of art glass, beaten metal sculptures, and smooth carved wood and stone things. We will have a special display area with magnifying glasses for pendants and teeny-tiny little sculptures. Maybe we'll have a little pretend kitchen set up with the really pretty wooden cutting boards and spoons, and the nice ceramic dishes, and like that. There will probably not be a lot of traditional paintings or photographs; my tastes in terms of things that hang on the wall tend a lot more towards mixed-media work. However, I do see some very nice large-format paintings at the Art and Foodstuff Festivals now and then, so if I've got museum-scale walls to work with I would probably wind up spending some of the budget on gloriously intense abstracts, and maybe the occasional landscape or nature close-up if it really fits with the mission. Yeah, maybe some Bay Area landscape photography every now and then, that'd work nicely. And then I would entertain myself coming up with different coherent organizing themes to fill the museum space with, that let the things on display express some kind of unifying concept without being overwhelmingly the same in style or color or mood or material.
Yay imaginary museum. Planning to haul art off to my imaginary museum may be just as much fun as planning to haul furniture and architectural details off to my imaginary castle. Hey -- I can put the imaginary museum in the imaginary castle! This explains what to do with all those extra rooms I needed to make the external scale sufficiently imposing! Huzzah!