In writing, how do you show internal dialogue?
You actually mean thought; dialogue is something between one or more people/characters.
There are a number of answers listed. Some have good information while others have not so good information. The one thing NOT to do is put quotes around internal thought.
As an editor, I cringe at the different ways authors try to invent to show internal thought. When in first or second person, all narrative is internal thought so there is no need to give explanations. But when you use 3rd person, you have to understand how close the narrative is to the main or POV character (unless you are writing omnipotent POV, which can be confusing to readers).
The usual way writers show internal thought of a character is to use italics. This is considered the most universal way to do so. There is no need to use the thought tag for all readers understand this type of style. So go ahead and use it for your manuscript.
The other method is very difficult for some writers to grasp. It is being very direct with the narrative, but not using italics or tags. Think of the following paragraph:
Jason stared out at the rolling sea. I really hope I don’t get sick. But he did.
In the above example, it is easy for any reader to know that the second sentence is internal thought, not dialogue or narrative. Not all internal thought begins with the work “I,” so don’t fall into that trap.
No matter what, you need to research and rewrite using test readers or critiques as you polish your work.