These two are easily confused. I had four semesters of Latin in college and I still have to remind myself of correct usage. Although both phrases are originally Latin, they are part of the English language, and thus do not require italicizing like we so often do with foreign words or phrases.
In Latin, i.e. stands for id est, while e.g. stands for exempli gratia. The former means “in other words,” while the latter means “for example.”
I see the two confused–most often i.e. going in place of e.g. erroneously. Only one of the two phrases actually contains the root for “example.” If you’re fishing for the right phrase to use in order to provide an example, e.g. is the way to go.
Ah, the quotation mark. It takes many forms…in the air, overused, underused, absent, and just plain wrong. Some use it for “emphasis.” (Hint: don’t.) Some put quotation marks common phrases in signs or marketing collateral. The short answer on all this is, of course, that quotation marks are marks for notating quotations. Not for emphasis. We might use them to set off a phrase someone said or their misuse of a phrase (such as a liar’s version of the “truth,” for example). But otherwise, they only work for quotes.
But within that, how are they treated? In American English, double quotations go around an actual quote. Punctuation (periods, commas, dashes, etc) go inside. It’s different in British English, but that’s beyond the point. When in doubt, consult this APA table.
There is an argument for logical punctuation, but as an old English major having two major publication styles chiseled into my head, this sort of thing makes my skin crawl.