Question: What is the best way to write out character conversations without the over use of quotation marks?Answer:
I don't think you can overuse quotation marks, unless by some strange reason you are using them where they do not belong.
It is true that some authors prefer not to use quotation marks, often from the belief that they can be an unnecessary distraction from the dialogue. They prefer a cleaner look to a page. You will see this style most commonly in literary fiction.
The downside is that, without quotation marks, readers have to work a little harder to decipher what is dialogue and what is narration. How much of a labour this is depends on your style. While some readers don't mind the extra work, and actually appreciate writers who have a non-standard style, others find it a distraction from the story.
You have to know your readers to have a sense of what they will find more distracting. However, the standard practice for most literature, fiction or nonfiction, and especially popular, commercial, or genre fiction is to use quotation marks around dialogue. Readers are used to relying on quotation marks to separate dialogue from narration, and because of that the marks themselves are largely invisible, much like the verb "said."