English does contain clues as to where cases might be used when a sentence is translated into Russian. Ex:

Replace the word with first person pronoun. If it's "I", it's nominative. If it's "me", it's accusative or one of the other cases.
If it's "to [noun]", chances are it's dative. (Also if "to" could be inserted into the sentence and it would still make sense, e.g. "give me" = "give to me")
If it's "of [noun]", chances are it's genitive.
If it's "with [noun]", chances are it's instrumental.
If it's "in [noun]", "on [noun]" or "about [noun]", chances are it's prepositional.
If it's none of these things, chances are it's accusative. Also bear in mind that whilst "to" = dative and "in/on" = prepositional, "into/onto" = accusative.

These are only very rough guidelines and there are many exceptions (and probably other rules, too). For example, "I am talking to you" = «я говорю с тобой», and «тобой» is instrumental, not dative.