I can't seem to figure out how to apply both an Umlaut (¨) and the Grave (`) to the the letter 'u'. I'll type one, then the other, then the letter, but only the latter accent is applied, with the former just left out to dry. Any ideas?
There are basically two kinds of Unicode characters with diacritics:
Key combinations like ⌥u just enter a dead key state where the next key press inserts some precomposed character. ⌥⇧u would insert a combining diaresis that in turn could be followed by another combining character.
Some precomposed characters with multiple diacritics on the U.S. Extended layout:
To enter some common diacritics almost everywhere in OSX, press the "ALT-combining-char" and then the key you wish to combine the diacritic with.
i.e. the following key combining combos can give you umlaut, grave, ring, acute, and tilde
NOTE: this doesn't work in Terminal.app if you have enabled the option "Use Option as Meta Key" in Terminal Preferences.
You can also use international input method as described below, to get access to many more diacritics.
Choose "U.S. International - PC" from System Preferences | Language & Text | Input Sources
Then type e.g.: apostrophe, grave, quotation, caret, tilde, double-quote and then your combining character to get the custom character. To get just the combining character alone, just press "space" after the character.