Long stuff ahead. I'm laying out the ground rules with examples for the uses of commas, parentheses, and dashes. This is a lot of stuff, but it's the best way I can think to explain when to use which, as it's a big question.
Your example sentence, "He spent several days wandering and collecting flowers (red and blue), and analyzed the results to prove his point," isn't a very good one. Let's get that out of the way first. It's much easier and generally more appropriate to punctuate sentences when adjectives come before nouns, as with, "He spent several days wandering and collecting red and blue flowers, and he analyzed the results to prove his point."
Now, with that out of the way, let's figure out commas and parentheses.
The comma (,) is used for many different things in English, and it can be difficult to remember them all.
- Use commas to separate independent clauses in sentences where those clauses are joined with a conjunction. For example, "This is one independent clause, and this is another independent clause." Note that both clauses have a subject and a verb and could stand as complete sentences by themselves.
- If your sentence starts with an introductory word or phrase, you might need a comma. Use commas if the introductory element is long. If the introductory element is only a few words, such as "therefore", "meanwhile", or "still", the comma is optional. When in doubt, it's acceptable to use a comma for all introductory phrases, so if you're not sure, use it.
- Items in lists of three or more require commas. "My grocery list includes apples, milk, and hamburgers." The comma on the final element of the list (in this case "milk") is optional. Like with introductory words and phrases, when in doubt, use it, as omitting it can change the meaning of your sentence. "We invited the police officers, James, and Julia," vs., "We invited the police officers, James and Julia." In the former example, police officers, James, and Julia were all invited. In the latter example, James and Julia are police officers.
- Like lists, nouns that are modified by three or more adjectives require commas between them. For example, "Jane is an intelligent, beautiful, and kind young woman."
- When you use words or phrases in the middle of a sentence that require a pause and aren't required by the sentence, use a comma. For example, "I sat, pensive, by the window." Do not use commas to separate subordinate clauses or phrases, such as with, "I didn't know that you were an Olympian!" and, "That he was an Olympian surprised me."
- As you've hopefully picked up, you need a comma to separate dialogue from non-dialogue. I need a comma before I start the quote, "This is my quote."
- Also, if you're going to continue your sentence at the end of a quoted sentence that should normally end with a period, replace the period with a comma. "This is my quote," is an example of a quoted sentence.
- Use commas to set off coordinate elements coming near the end of the sentence. For example, "He's only ignorant, not stupid," and, "There is no one alive who is youer than you, except you!"
Doozy. There are other places where commas should be used, but those you need to learn to recognize intuitively. In general, use a comma whenever one is needed to enforce effective communication and make the meaning of your sentence clearer, but don't overdo it! Only use a comma if it fits a rule or if omitting the comma makes your sentence mean something different.
Parentheses, fortunately, are far easier.
- Use parentheses to separate information that serves a clarifying role or is included out of context as an aside. For example, 'The comma on the final element of this list (in this case "milk") is optional.'
- Only use punctuation inside parentheses if the entire parenthetical is a complete sentence that is not contained within another sentence (in other words, could stand on its own). (This and the previous sentence both contain examples.)
That's really it for parentheses. They're a powerful tool, and it's easy to remember how to use them because there aren't many rules.
For dashes, things get a little trickier. You rarely need dashes.
- Use a dash when you need a long interruption to a sentence--longer than a comma can provide.
- Dashes can also help you emphasize a point. "You are one of lucky ones--the only one, in fact--on the closing shift tonight."
- A dash at the end of the sentence indicates broken dialogue or a broken thought process, as though the dialogue or thought has been interrupted. 'She began to shout, "Don't you dare --" but grew suddenly quite when she realized it was already too late.' Note that sentences broken in this way do not use punctuation at the end.
- Lastly, a dash can illustrate a sudden shift or change in thought, but this usage is very similar to the third bullet point about broken thought processes. "Take your sister with you, and -- Oh shoot, I forgot to get milk at the grocery store!"