This is something I honestly didn't understand for a long time, until sometime this year, actually. I don't blame you- it took me a while to catch on (plus ELA class), so I'm here to help. 

I'm going to give you a few correctly punctuated and spaced examples. Try to find patterns. There is a bit of sense to it, believe it or not. 
*Hint: Look at capitalization after the ", and what ends the sentence.

1. "Bobert, please help us," pleaded Sally.

2. Bob strolled through the garden, whispering, "Oh, beautiful cherry blossoms."

3. "Well, of course, he wouldn't do that," Sally explained, "but it's not like he understands."

Let's look at example #1. ["Bobert, please help us," pleaded Sally.] 

The first word is capitalized- it is a proper noun and the start of a sentence.
Then, after the word "us" is a comma,  because the 'tag' is after the quotation marks.
The tag is the part that tells us who is saying it and how they are saying it (he said, she cried, he yelled, she exclaimed, etc).
If the tag was before what was being said/the quotation marks, then the quotation marks would end with a period [with exclamation points, question marks, and dashes, simply put a comma after your tag before you have your quotations]. 

Now, a look at example 2. [Bob strolled through the garden, whispering, "Oh, beautiful cherry blossoms."]

There is a comma after the tag (whispering), right before the quotation marks.
Again, the first word is capitalized inside the quotation marks.
The reason why we don't have a period before the quotation marks is because we think of the quotations and the tag as a part of the same sentence (or at least that's an excuse/reason for it).
Again, we have a period at the end of the quotations, INSIDE the quotations.
All punctuation belongs in the quotations. If you are quoting something (like from an article) that doesn't have punctuation at the end of it for whatever reason, add a comma/period or leave it blank. If you are just quoting it and speaking about it like this-

"The cat's out of the bag" is such an overused expression.

-then leave out the comma/period. If you are saying it like someone is saying it, then add a comma or period depending on where the tag is. 

Onto example 3. [ "Well, of course, he wouldn't do that," Sally explained, "but it's not like he understands."]

There's a comma after 'that,' and Sally (of course) is capitalized.
If we replaced 'Sally' with 'she', we wouldn't capitalize 'she' because it is a part of the same sentence. The next word in the dialogue is 'but' and is not capitalized.  
Also, the period is at the end and inside the quotations.

Wow, that's a lot of information. I didn't cover everything, I don't think, but it's already a lot to take in. Hopefully this helps you guys!

[I think I'll be doing parenthesis next! Perhaps I'll make a part two for this about spacing and such. Or maybe I'll do : and ;! And !? vs ?? vs !!/tags and making a thing about punctuation...or maybe similar words & their differences. Vote below what you want to see!] 

~Lion