Keep in mind you're getting one side of the picture from Hazlitt. Hazlitt's a libertarian--he thinks less government is better. So he's not very prone to give you much evidence that sometimes, governments can do desirable things, just because that's not his bias.

The best advice I can give you is to try and separate facts from judgments. Take the minimum wage. It's well understood that minimum wage laws cause unemployment. It doesn't necessarily follow from this that you don't need a minimum wage, though. Without a minimum wage, you might have twice as much unskilled labor working for half the pay, but all of them are below subsistence level and require government assistance. Whereas a higher minimum wage means some people can be dropped from the rolls entirely while others have to collect more. Or maybe you change the welfare system. It's a judgment call which tradeoffs you want to make, just like engineering. There are obvious things (command economies are pretty crap, just like bubble sort is a crap algorithm) but there are more subtle things (like time/space tradeoffs that dictate different algorithms for different applications) that depend on what you're trying to do.

Likewise, consider health care. Some people think it's a moral imperative that a wealthy society provide health care to everyone. Economics asks some tough questions--at what cost, what level of care do we provide--but it also provides answers--what are different ways of doing this, and what are the cost-benefits?

Economics at its best is a science. It doesn't dictate your goals to you, it just warns you of some natural constraints. You might say, "I believe in minimal taxes and don't care about poor people who need housing and health care but can't afford it." Well, economics tells you that's very feasible. You might say, "I believe in minimal taxes and a robust welfare state." Economics tells you there's no free lunch. You might say, "I believe everyone should have health insurance that meets these requirements." Well, economics says, you could give everyone health insurance, or you could have a market in health insurance and give people vouchers to pay for it, or do any other thing you can think of, but here are the effects and tradeoffs of those policies.

In terms of what in economics to study? Microeconomics seems better grounded than macro. Behavioral economics seems promising. Austrian is a mixed bag--there's some stuff that promotes good economic reasoning but then there's also gigantic apologias for anarcho-capitalism based upon some moralistic conception of natural law that totally mix up the science of causes and effects with the moral aspect of goals and intentions.

I find that separating causes and effects from goals and intentions is a big problem not only in economics but also in ecology. There's a difference between saying "I want to sustainably maintain an environment that is good for human life" and "I want to preserve the environment mostly the way it is without human tampering", and an environmental scientist will recommend different policies based upon which goal she implicitly has in mind. Maybe some endangered species can go extinct if there's a better tradeoff for humans, or maybe not. Likewise with economics: there's no such thing as "economics says policy X is bad", only "economics says policy X has effects Y".