The following is a paraphrased topic request from a loyal reader:
Grammar Cat, you are so awesome with grammar. I think you are the paramount source of cat-related grammar on the internet. Please tell me about how quotation marks work with other punctuation.
Well, grotesquely paraphrased reader, the good news is that this doesn’t have to get too complicated. The bad news is that you will have to do some memorization. The first thing you must do is abandon almost every fundamental need for a consistent rule. The second thing you must do is blame the British for the madness that is American quotation punctuation.

“Typical.”
British rules tend to follow a consistent string of logic. However, Americans at some point decided to do our punctuation a bit differently, just to be…well, different. The most obvious difference between British and American rules is that standard quotation marks for the Brits are single marks like so: ‘I do say, Sir Purrsalot, this custard is brilliant!’ while Americans prefer, “My cat ate a cheeseburger the other day and everyone thought it was funny, but I have no idea why.” But we’re not British, right?

Darn tootin’!
I’m pretty sure quite a few people had to die to make it this way, so let’s not disgrace their memory by screwing with punctuation.
PUNCTUATION BEFORE QUOTATION MARKS:
Punctuation that helps introduce a quotation will always go before the opening quotation mark. Whether it’s a comma, period, colon, semicolon, em dash, or whatever other harebrained thing you can think up, it will always be correct with the punctuation mark preceding the opening quotation mark.
I swear my cat just said, “I love you, Daddy.”
The cat’s eyes bore into me until I cracked. “Okay, fine! I did it! I spilled your milk saucer!”
He then understood what was happening: “Dude, I think my cat is on drugs.”

“Sorry, but I ate all of the green stuff that you always hide in your sock drawer when your mother visits.”
PUNCTUATION AT THE END OF A QUOTATION:
This is really the main event, right? This is why you’re here.
Periods and Commas
Here’s a simple thing to remember:
Periods always go inside the closing quotation mark.
And guess what.
Commas always go inside the closing quotation mark.
Repeat after me: Periods and commas always go inside the closing quotation mark.
“So many hairballs, but so little time,” the cat said sultrily.
“I think I’ve seen that cat kill a man before.” Steve’s fearful eyes never left Fluffy’s as he backed away.
The talking cat’s first words were, “milk,” “love me,” and “okay, we’re done here.”
Colons and Semicolons
Here’s another simple thing to remember:
Semicolons always go outside the closing quotation mark.
Same goes for colons.
Colons always go outside the closing quotation mark.
Now repeat after me: Colons and semicolons always go outside the closing quotation mark.
I should have said, “I’m sorry your cat is a communist”; instead I read him Das Kapital.
I noticed something strange while reading William Carlos Williams’s “Poem (As the cat)”: I am a cat.

“It all makes sense now.”
Exclamation Points and Question Marks
Now let’s get into some grayer territory and learn about question marks and exclamation points.
Question marks and exclamation points can go inside or outside of the closing quotation mark.
The way to determine where to place your ? or ! depends on the context of the sentence. If your quotation is part of a larger question or exclamation, then the ? or ! it will go after the closing quotation mark, and no other punctuation is needed before the closing quotation.
For example:
Would you rather spend a day in a freezer with a wolverine or spend an entire day listening to “Honky Cat”?
To quote Socrates, you idiot, “Kittens don’t eat broccoli”!
BUT
“Do you have any Grey Poupon?”

“Why, I never!” the cat said when the waiter admitted he had no Grey Poupon.
As you can tell from the last two examples, when the ? or ! is actually part of the quotation, it comes before the closing quotation mark.
So yes, there’s a bit of thinking when it comes to the question mark and exclamation point. Sorry about all that.
QUOTATIONS WITHIN QUOTATIONS:
Prepare to have your mind stretched to its limits!

“A quotation within a quotation within a quotation within…”
When a quote has a quote within it, Americans use a single quotation mark to denote the secondary quotation. So say you’re writing about your petty friend imitating his cat. He believes the cat said, “Give me some food and shove off.” Now, since we only quoted the cat in that example, the speech goes in double quotes. But when we quote our petty friend’s rendition, then suddenly the cat’s words are embedded within our friend’s speech, making it something like,
“My cat is such a mooch. He just walks up to me and says, ‘Give me some food and shove off!’ and then walks away.”
Notice that the embedded quote of the cat is punctuated with the exclamation point just like a double quotation would be. And that’s the trick. Quotes within quotes are treated just like regular quotations. The only difference is in the single or double quotation mark.
For those of you who never know when enough is enough and are asking, “But what if someone is quoting the Grammar Cat’s example of a quote within a quote? How does one indicate a triple-layer quote?” Okay. Here’s how it would go: The Grammar Cat says, “…then suddenly the cat’s words are embedded within our friend’s speech, making it something like, ‘My cat is such a mooch. He just walks up to me and says, “Give me some food and shove off!” and then walks away.'” Always start with a “ and then go to a ‘ then back to a “ and so on and so forth.
However, here’s an easier rule:
If you find yourself using a triple-layer or (dog forbid) quadruple-layer quote, find a way to rework the sentence so that you don’t have to do that.
Readers will thank you.
Also, when there’s a single quote and double quote next to each other, it looks terrible. CMoS acknowledges this fact and allows for a space to be placed between them for readability’s sake, if one so chooses.
QUOTATION MARKS FOR EMPHASIS:
This last tidbit on quotation marks feels more like a public service announcement than a grammar lesson. There’s this weird thing that people do nowadays when they have too limited a vocabulary to use connotation to their advantage and think that for some reason italics, underlining, and/or bolding doesn’t provide enough emphasis on a particular word. This terrible thing is where people use quotation marks to add emphasis. This is not a function of the quotation mark. Here is an example of how quotation marks are used incorrectly:

The above makes no sense. Is there really a dog? Is there just the concept of dogness happening? Is there just a cat whose name is Dog?

“Beware me!”
The only way that random graffiti would make any sense is if saying the word “dog” was something to beware.
What’s even worse is that now there are people who see others using quotation marks to add emphasis and react with, “Oh, that’s just silly. Everyone knows it’s the single quotation mark that’s used to add emphasis.” WRONG. THAT IS SO WRONG. That makes even less sense, because there is only one use for single quotation marks, and that is listed above under Quotations Within Quotations.
Here are some examples of incorrect use of quotations, both single and double:
My cat stopped eating his “so-called” food.
How many more times until my cat realizes I ‘don’t’ want him to shed?
Just writing those examples is making my skin crawl, and yet it happens everywhere. I almost feel bad for pointing it out, because now you’ll notice it everywhere.

Sometimes I “use” quotation marks because it makes it seem like I’m not the one “responsible” for my stupid words, but that I’m just “quoting someone else.”
It’s unclear whether people started abusing quotation marks to avoid taking personal responsibility for their own words, instead trying to pass the buck to some other unnamed source, but that seems pretty typical for the internet.
Since it only seems right to end the quotation mark post with a quotation, I’ll leave everyone with the words of the great Theodor Geisel. “Be who you are and say what you feel, because those who mind don’t matter, and those who matter don’t mind.” Grammar Cat can’t be sure, but we think he wrote this in regard to the misuse of quotation marks.