I don't think that there is a generic answer that will apply to this question across all goods. The answer depends on how the related businesses work, how much insight you have into the true value of the goods, and probably other things. Your car example is a good one that shows multiple options - There are dealers who will buy as a single transaction, sell as a single transaction, or do a simultaneous sell with trade-in. I had a hot tub once, on the other hand, where I could find people who would do a trade-in, but there was no dealer who would just buy my used tub. There's not much parallel between the car and the tub because the options available are very different.
To the extent that there is a generic answer, I generally agree with the point in @keshlam's answer about trying to avoid entrapment, but I take a slightly different view. If you want to get your best deal, you need to have an idea going into the process of what you want in net and keep focused on meeting your goal. If for some reason, it's convenient for the dealer to "move money around" between the new car and the trade-in, I'm ok with that as long as I'm getting what I want out of the deal. If possible, I prefer to deal with both transactions at once because it's simpler. At the same time, I'm willing to remove the trade-in from the deal if I'm not getting what I want. (Threatening to do so can also give you some information about where the dealer really puts the value between the new car and trade-in since, if you threaten to pull the trade-in, the price on the car will probably change in response.)
