Correct.
Coffee shops are everywhere and so are those take-out cups. They even make reusable cups that look like the throw-away kind.
If yours is reusable, wash it and reuse it. But if it is cardboard, plastic, plastic foam, or any other "disposable" substance, dispose of it and the plastic lid in the trash, not the recycling bin. That includes the sleeve that keeps your fingers from being burned.
Fast-food soda cups, plastic lids, convenience-store cups and similar products should not be recycled. For starters, it would be hard to get them clean enough to recycle, but don't even try.
There are some fast-food cups that are a similar plastic to the clamshell containers you buy fruit in at the grocery store. If you take them home and wash them, you can recycle them, but not the lids.
