Will I set aside part of my paycheck for health insurance, retirement savings, or both? Why or why not?

If you live in a country that provides either health insurance or retirement savings, or both, you don’t need to worry about one of them or both of them.

On the other hand, if you live in a country that doesn’t provide for any of these, I recommend looking in to what your company can offer you.
Most big companies give you a fair amount of money when you quit your job because of old age. If this is the case, health insurance should come first.
Very big companies might give you both health insurance and retirement savings. In this case don’t worry about anything.

Even if these are the facts, I would say that a different bank account for health insurance would be useful. For this, there are a lot of things you could do.

The best thing to do in my view would be to be engaged in the stock market. You should invest money in stocks from small companies that have a good track record.
The next best thing to do would to open an automated savings accounts. This is a savings account that will automatically take out a fixed amount of money from your actual bank account. This is also good if you want to have retirement savings.

I hope this helped. :)