> I don't understand how idealism can be a complete theory of the world. What are the minds made of?

Ask an M-theorist "what are strings and branes made of?" They'd probably say something like "in M-theory, they are fundamental entities, they are not made out of anything more basic". Well, an idealist can say the exact same thing about minds. Different minds have different properties, just as different strings and branes can have different properties, but for both theories they are entities which cannot be reduced to anything more fundamental.

> What are the physical laws that govern their working?

From an idealist point of view, I'd say physical laws are regular patterns which exist in the experiences of minds. So, your question then becomes: What regular patterns in the experiences of minds govern the working of minds? The question doesn't make a lot of sense; the best sense I can make of it is – exactly the physical laws you are familiar with. They govern the workings of minds, for (as far as we know) minds always have experiences compatible with the truth of those laws.

> How do we know there are more kinds not just one, my own (and arrive at solipsism)?

I would say that the existence of other minds is an innate idea, an a priori concept. It is a belief which does not require any evidence to justify – because the entire idea that "beliefs require evidential justification" is part of an epistemic value system which has the existence of other minds as one of its unquestionable presuppositions.

> The Standard Model at least posits a simple world, made of a relative handful of basic elements that follow some simple laws. Minds are extremely complex objects, and it seems very hard to accept them as irreducible - especially if you chose to believe that other people and their minds exist as well.

But how are you defining "simple" and "complex" here? From another point of view, the physicalist world is enormously more complex than the idealist world. If we assume there are no omniscient minds, then the universe must contain a vast quantity of irreducibly complex information which no mind will ever know – I've read one physicist suggesting that the information-theoretic complexity of the observable universe is on the order of 10^120 bits – whereas a (non-theistic) idealist can say that no mind will ever know the vast majority of that information, and if no mind ever knows that information it never actually exists, so in a (non-theistic) idealist worldview the infomation-theoretic complexity of the observable universe is likely to be substantially less, meaning the universe is substantially simpler. (Whether the same would hold in a theistic idealist worldview is a more complicated question which I suggest we leave to the side.)

Now, you might object to how I am defining "simple" and "complex" here; but the terms have no precise and unique definition – and even if some theory is simpler than another by some specific definition of "simple" and "complex", why should we care? Even if preferring theories which are "simpler" in some specific way turns out to be the most productive approach to the natural sciences (compared to preferring theories which are "simpler" in some other way) – why should we assume that the same must be true for other disciplines with quite distinct methodologies, such as philosophy?

> Having read a bit about Berkeley's idealism from the Stanford Encyclopedia of Philosophy [0], to me it seems that it is simply a re-stating of mind-matter dualism with other terms

I'm sympathetic to Berkeley-style idealism, but wouldn't defend every detail of Berkeley's views. Some idealisms do contain some degree of dualism – the mind itself vs the contents of the mind, experiencer-vs-experiences, my own mind vs the minds of others – but I don't agree that any of those dualisms can simply be reduced to or identified with the dualism of mind-vs-matter – and there is no single mind-matter dualism, since mind-matter dualism exists in multiple varieties: substance dualism, property dualism, hylemorphic dualism, etc, which disagree with each other about what mind and matter are and how they are related to each other.

> But then what is causing the ideas? According to him, a special spirit called God

This is one element of Berkeley's views I definitely disagree with. I think he is too quick to resort to God as an answer to questions which I believe can be answered without invoking the concept of a divine being. (You can be an idealist without being a theist – McTaggart is a famous example of an atheist idealist – and even a theistic idealist might prefer to invoke the concept of God more sparingly than Berkeley does.) In my view, causation is simply another kind of pattern which exists in the experiences of minds – to ask "what causes experiences" becomes equivalent to "what experience-patterns of the particular type we call 'causation' do experiences participate in?" – the obvious response to which is "the answer will be different for every experience". An idealist can provide a theory of causation without bringing God into it.