Generally, my understanding of the use of dative and accusative is a bit opaque. I am getting better at deciphering which prepositions call for which case. I have also seen that motion is another indicator. If I am not clear, however, is there a resource which I can use to look up a particular word? (I use dict.cc a lot and I don't think it specifies.) Usually I Google "German dative verbs" and use CTRL+F to search for the verb, then assume that if it's not found, it's accusative.
Half of the answer has been given (
Here two other points to consider.
First, the answer is a little grammar as well. Not all verbs have a fixed case; for instance waschen:
Ich wasche etwas Akk..
Ich wasche mir Dat. die Hände.
So you might have to do the analysis by yourself.
Second, etw and sich are also an indicators of dative and accusative. You might also find in dictionaries then
By the way, don't neglect genitive-objects ;) Not the best way to perform the search, but this one gives you lot of them. Their register is usually gehoben.
jdn to indicate accusative (for example, see dict.cc on "lehren") and
jdmfor dative (see dict.cc on "ausweichen", e.g.). But the indicator may be missing, cf. dict.cc on "verzeihen". Often, the example sentences are giving a hint which grammatical case to use. There's also an explanation of the abbreviations used (but this tells only that
jdn means "jemanden", but doesn't explain why those indicators are added to the verbs).
You can also look up the verb in the free Pons online dictionary. (Pons is one of the better known publishers of dictionaries in Germany). It will list many (but not all) verb patterns, with "jdn/jdm" or "Akk./Dat." etc. for the cases. I recommend learning the patterns and the associated meaninings instead of just the verb (in the same way English students have to learn "to put up with s.th." instead of just "put", you learn "sich (Akk) mit jdm/etw abfinden" instead of "abfinden").
For prepositions that don't belong to a specific verb pattern, mein-deutschbuch has some good lists.