Last time I checked we (America) have to pay for that, not them?
Yes, the business that is importing the goods from the country pays the tariff, and then the costs get passed onto the customer, because the product is now more expensive to make. It is not an expense the country being tariffed pays.
In some scenarios, like mentioned above where oil is shipped to the United States to be refined and then shipped back, if the other country imposes a retaliatory tariff, then that final product is going to go through multiple layers of taxation and prices will skyrocket.
In theory, modest tariffs can help protect limited industries that it might make sense to keep robust in the country. Like corn, for example, since the United States is such a huge producer (we make about one-third of the world's corn, period), there could be an argument made for imposing a modest tariff on importing corn from other countries, to encourage American businesses to support American industries, reduce carbon footprint of importing from overseas vs. two states away, etc.
But these across-the-board tariffs are nonsensical. We import 30% of our lumber, 90-97% of which comes from Canada in any given year. We can't suddenly create old-growth forests, and have a bunch of entrepreneurs start up lumber businesses, train staff, build out manufacturing facilities, etc. over the weekend. The only end result of that is massive price hikes on anything that needs lumber in its processes, which includes not just obvious things like building houses or furniture, but any business that might need lumber for something, even if they sell something like, IDK, stickers, now has a portion of their business that is significantly more expensive, and that cost gets passed onto consumers. And not even purely from a greed perspective, if the choice is "raise prices or go out of business," most businesses will pick the former first.
Sorry, went on a slight "OMG, let me explain this" rant.

My degree is in economics, and not you, but seeing people in general who are gung-ho for Trump's tariffs without understanding what he's actually saying is so aggravating to me.