Learn these prepositions, and learn 'em good. Know that you can count on these two groups of prepositions. They stick to their guns. They are dependable. They won't let you down. All their objects come out inflected as though they were being used as direct objects (for the DOGFU group) or indirect objects (the datives, further below). What's all this Dative and Accusative mess? Ah, you have not yet visited the Shrine of the Holy NADGe.

We know what direct and indirect objects are, right? The subject of a sentence acts directly on the direct object via the verb, and indirectly on the indirect object (if there even is one). This happens quite naturally without prepositions getting in the way, as in Ich kaufe dir das Buch.

subject verb indirect object direct object indirect object
Ich kaufe das Buch.
I'm buying the book.
What gets bought? The book.
Ich kaufe dir das Buch.
I'm buying you the book.
Alternatively
I'm buying the book for you.

I am the subject, the book is the direct object (that which is bought), and you are the indirect object (for whom the book is bought). Oh, look at that — in English we also use prepositions to indicate an indirect object: for whom. Never needed to think about it that way before learning German!

But in German, there are at least two circumstances under which the direct/indirect objectiveness is overruled. One of them is via prepositions. When a preposition acts upon a noun in a prepositional phrase, that noun becomes its object. The preposition simulates, for want of a better term, a particular grammatical case for its object. The DOGFU list forces its object into the Accusative case (meaning those objects appear as though they were being used as direct objects even though they're not — they're just being used in a prepositional phrase). The list of Dative prepositions does what you suspect it does.

Accusatives

durch
through
ohne
without
gegen
against
für
for
um
around

Fortunately, they line up nicely into a goofy mnemonic. Much better than our next group, which doesn't have a catchy name. I recommend rote memorization here, perhaps in an angry little Teutonic chant. Aus! Außer! Bei-Mit-Nach! Seit-Von-Zu! Or go totally elegant, like an ostrich1 in a ballerina tutu.

Big thanks to the esteemed members of the English-Speaking Music Ensemble e.V. for teaching us the Blue Dative Danube Eselsbrücke thingy!

/assets/blue-danube-datives.mp3

Datives

aus
out, out of, shut off
außer
excepting, outside
bei
with, by, in case of
mit
with, at (the age of)
nach
after
seit
since
von
from, by (authorship)
zu
to, closed

Try to internalize DOGFU and the Datives™ (that's my grammatical cover band name). It's nice to have hard and fast rules in languages (how many can you think of in English?). German, for all its irregularities, has a few areas where you can count on it to come through for you in a pinch. Take advantage of them!

1. Strauß! Get it!?↑