I was looking through the points made by Berkeley and I fail to see how he arrives at my not having an idea of an object existing unconceived. — Darkneos

His argument is that all ideas of mind-independent objects smuggle in a perspective. You imagine a tree as if you were looking at the tree from some angle. What you can't do is imagine the tree from no perspective. He also rejects abstractions as real entities, so therefore abstract notions (like mathematical models) of material objects don't work either.

Keep in mind the above is an attack on indirect realism, where material objects are inferred from mental ideas/representations. He first has to go through arguments disproving direct realism, because obviously if we're directly aware of material objects, then Berkley's idealism doesn't get off the ground.

As for solipsism, it's a legitimate concern in my view, but I guess minds are given in Berkley's philosophy. Also, so is God, who is required to keep ideas persistent when we're not perceiving them.

Berkley's idealism is just one form of idealism. There are others with their own strengths and weaknesses. And he wasn't the first. Idealism can be found in ancient philosophy. The motivating factor was the problem of perception, and later on, the varying ways humans categorize and make sense of the world.

The problem I have with idealism is how can the world be made of ideas when we haven't even learned them yet?

I also find that most idealists posit a "mind at large" that holds perceptions, for which I can't honestly see anyone actually believing it.