There is conscious activity (thinking, feeling, sensing, hallucinating, dreaming, and so on). Idealism is the claim that there is only conscious activity. Given idealism, one then has to provide a scenario or two that would explain why it seems to us that there is non-conscious activity, like water flowing down a river. One scenario is to say that we could all be sharing a dream. Another is to say we are avatars in a virtual-reality-like simulation, albeit one produced by superior spirit beings, not running on a computer. It should be noted that these scenarios are not additional assumptions, rather just ways to imagine how it could be that water flowing down a river could be understood as being within consciousness. On the other hand, those who assume there is something other than conscious activity are faced with either the intractable problem of how conscious and non-conscious actvity interact (dualism) or the hard problem of consciousness (materialists). So idealism simplifies by not having an intractable problem.
But that's just the start. Other ontologies have a language/reality problem: there is reality, and then there is language about reality, with the issue of how well language can describe reality, indeed, whether it can at all. But with idealism, language is simply more conscious activity. With other ontologies one has to worry about the fact that the map is not the territory. But with idealism, a territory is a highly detailed thought construct, while a map of it is simply the same construct without all the details. Of course, a map may leave out some vital features, and so be a faulty map, but to say "the map is not the territory" is no more informative than saying that a design of a house is not a house.
(That there is no language/reality distinction in idealism also makes idealism immune from post-modern critique. See "Idealism as a Response to "Postmodernism"" for more on this.)
Then there is the reality/appearance distinction. With idealism, all appearances are real. End of story. Well, one might point out that when we look at water flowing down a river, that that appears to us as non-conscious activity. In response, I would say that it is not a false appearance, rather, it is a false belief about what appears. We are not cognizant that it is a dream river (according to one scenario) or that it is a representation of the thought of highly advanced spiritual beings (according to another). A simple way to grasp this is to imagine someone who has never seen or heard of writing of any sort, then presented with a scrap of paper with some writing on it. That person will not be cognizant that the words on the paper mean something -- it will just look like random marks.
Next up, the ontology/epistemology distinction. With idealism, to modify Berkeley, to be is to be known, and so a theory of knowledge is at the same time a theory of being. So no distinction.
Similarly, there is no ontology/logic distinction. Taking logic in a more general sense, as the study of patterns of thinking, and since thinking is conscious activity, the study of logic is again the study of being.
Although this is controversial among idealists, I would say that with idealism there is no ground/grounded distinction. There is conscious activity, and there are particular conscious acts. But that just means that the term "conscious activity" is the generic term for all those acts, whether of God or human. Of course there are conscious acts which depend on prior conscious acts, and so one might pursue this to claim that there is (or was) a Primal Conscious Act which is the ground for all other conscious acts. Or not, that is, there may be an eternal generation of spontaneous conscious acts, some of which coalesce as systems of causation, while others don't.
Lastly, though this one requires a logical shift, many of the traditional bugaboos of metaphysics such as the problems of one/many, permanence/change, continuity/discreteness, eternity/time, and being/becoming, can be understood as features of every conscious act. This means they can be seen to no longer be problems arising from what we experience to seeing both poles of these polarities as necessary for any experience. The ground is not one pole or the other, but both in a nondual relation. See the Tetralemmic Polarity essay for details. The shift is that, while what (in other ontologies) is seen as a problem of what is experienced (is it one or many, etc.), can now be seen as simply the essence of experiencing.
Of course, none of this "proves" idealism. It's just nice to sit back and watch most of the problems that other ontologies have to deal with evaporate.