Can an American go to Cuba?

Travel Info
My CWP teacher says that they can't, but he's been known to be full of crap before and I just wanted to make sure.

Travel Tips

You CAN go to Cuba. However, you need to get a permit from the US Treasury Department given that you have a justification. People who go there because of work, or study abroad programs (like a friend did), or other specific reasons listed in the Department of State website are JUSTIFIED reasons to go down there.
You can't just go as a tourist. The US won't allow it and you'll have problems at immigration checks at airports if you do. Penalty is fine and/or prison.

Others
you can go to cuba but it is at your own risk
Yes, but you have to go through Mexico
They of course go to cuba on vacation and look it , I dont see why they could not.
An American can't go to Cuba from the United States, but you could go to Cuba from another country. It is because the U.S. doesn't support the way they are ruled(tyranny).
The US says no but I have known people who have gone but from other countries, like flying in from other Caribbean or Central American countries. However I don't know what the US government officials would say when reviewing your passport when you re-enter the US.
There are two ways I am aware of.

1) Go from outside the US. For example, many go through Mexico. They just don't get their passports stamped.

2) Participate in an official trip involved with medical assistance. I know people who went as volunteers, and they aren't doctors or nurses. They just have to bring medicine or medical equipment with them as part of an official program.