The best way is to learn the complete verb patterns. So don't learn "geben", learn "jemandem (Dat) etwas (Akk) geben". Don't learn "abfinden", learn "sich mit etwas (Dat) abfinden" (to put up with s.th.), or "etwas (Akk) etwas (Akk) nennen", or "sich dessen (Gen) entsinnen". Also, learn prepositions together with the cases they take (von + Dat, in + Dat/Akk, etc.). A good dictionary, e.g. Pons (http://de.pons.com/), will have the verb patterns with their different meanings (often in abbreviated form, e.g. jdm = jemandem (Dat), jdn = jemanden (Akk) etc.)

Unfortunately, not really, but more or less, you could say that the direct object in English is represented by the Akkusativ and the indirect object by the Dativ, but of course, this is a rule of thumb. For example, when you have the question: what do you have/need/want/eat/see etc., these verbs would all require the Akk., but when you ask: to whom do you give/send etc. this, you need to use Dat. And then, sometimes it doesn't work, e.g. in English you help somebody, but in German 'helfen' requires Dat. Another thing are the questions 'wo?' and 'wohin?'. 'wo?' requires Dativ, 'wohin?' requires Akkusativ (with the exception of 'zu'+ Dat.).