In what ways should and shouldn't the government be involved in the economy?

Certain regulations are beneficial, specifically those that prevent monopolies and anti-competitive practices from taking root. Competition is the soul of what makes a free market effective, without it the free market system doesn't exist. Not only that, but certain sectors are necessary but unprofitable, or unethical to profit from. One example being police work. A profit driven police force is a corrupt police force. The social safety net is a other valuable tool that governments can exercise. Though some people dislike them, when efficiently executed, such safety nets do improve public welfare, safety, and economic mobility.

As far as what the government should not do, profit. A government run for profit is a government corrupted. Ideally they can just cover their expenses, with a small deficit or surplus outside of times of great need.