The two little dots one occasionally sees over a, u, and o is called the umlaut, as we all know. It represents a vowel change also known as umlaut. This vowel change often has grammatical significance. Frequently, for example, plural nouns will have the umlaut. Also, some forms of some verbs will have it, and others will not. The obvious word pair distinguished by the umlaut is schon already, vs. schön pretty, fine.
As far as pronouncing o- and u-umlaut, my best advice is Don’t take lessons from Wayne Newton. The next best advice is listening, over and over, to a recording spoken by a native speaker (that is, if you can’t find a native speaker to help you). The third best advice is try what the textbooks say: form your lips to pronounce o or u (as the case may be) and say German e (long or short) with o, or i (long or short) with u. For me, ö is the hardest to pronounce naturally. A-umlaut is the easiest: just pronounce the German e sound (long or short), no need to pucker or do anything fancy with your lips.
The two dots originally represented a shortcut way of writing ‘e’ after the vowel. Some proper names are still written that way: Goethe and Bonhoeffer, come naturally to mind. In a pinch, you can always use that as a way of writing; if you can’t figure out the keyboard shortcuts, you can always write fuer for für, for example.
English preserves a few forms that have grammatically significant vowel change: man and men, for example; or even woman and women, which has an unwritten vowel change in the first syllable (i.e., o to i).
Theological German/Theologisches Deutsch 
When I am typing in German, I always use the German keyboard layout, which means that the letters with umlauts are there without having to remember complicated keyboard shortcuts. You can access this on a PC by going from Start into the Control Panel, then Regional and Language Options, then clicking on the languages tab, then details. In the Installed services box, click add and add German (Germany) as an installed language. This will offer you the option of a German keyboard. Click OK all the way out. You will then find a new icon in the bar on the bottom RHS which says EN. Click on this to change to DE whenever you want to type German and again and select EN when you want to go back to English.
The only problem is that there are a few keys that have different characters mapped to them in the German keyboard, so you have to watch this. For example the ‘z’ and the ‘y’ are swapped and if you press the apostrophe key, you will get an ‘a’ with an umlaut on it.
You can display a keyboard on your screen to help you learn where the keys are. Hold down the Windows key and press U. Select the line that says the onscreen keyboard is off, then click Start. A keyboard will open up on the screen. Click on it and then change your language to DE. You will see where all the keys you want are. You can click on them with the mouse if you like (slow and tedious) or just move it out of the way and use it to find any the location of the keys you want on your normal keyboard.
Thanks Judy, great tip!
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An even better method is to use the ALT codes for the umlauts. Google german umlat codes. Much easier and no messing with keyboards.
For example ALT+0223 – ß Always use the “0” zero in the code.