The term for the class of verbs that are causing the OP problems are called prepositional verbs. These are verbs which take an object together with a preposition.
Example of a prepositional verb:
- Ich glaube an Gott. (Correct)
- Ich glaube Gott. (False, because the preposition is missing)
The motion verbs do not fall into this category because they do not need a preposition to specify its object.
Example of a non-prepositional verb:
- Ich binde das Pferd. (Correct, no preposition is needed)
There exist a rule for figuring out the case of non-prepositional verbs, but as the user Georges points out in his answer this rule is not so easy to apply as one might first think.
No, there are no such governing the case of prepositional verbs. According to Virginia Commonwealth University and University of Michigan you should learn the case by rote in each case.
But still, there are some tendencies which might help learners of German.
Quick'n'dirty suggestion for learners
The website www.deutsched.com notes in a grammar lesson that prepositional verbs with a two-way preposition usually take the accusative. This is true for all the two-way prepositions except vor which usually take the dative. Therefore I suggest:
When unsure about the case of a prepositional verb guess the accusative case except if the preposition is vor, then you should guess dative.
More elaborate info for more ambitious learners
The website vistawide.com go into the details of many prepositions. I simply copy-paste the parts relevant for this question. Visit the webpage for loads of good examples
An In verb + preposition idioms, the two-way preposition an is used more frequently with the dative case than with the accusative case. Most of the an + accusative phrases refer to mental processes. The preposition in an + dative idioms often means in connection with, with respect to.
Auf In verb + preposition idioms, the two-way preposition auf is almost always used with the accusative. In the few instances where auf occurs with the dative, it indicates an enduring position, a lack of movement. Auf is the most commonly used preposition in verb + preposition idioms.
In In most verb + preposition combinations, the two-way preposition in is used with the accusative.
Über In verb + preposition idioms, the two-way preposition über is always used with the accusative case. In a number of idioms, über means about.
Vor The two-way preposition vor is always used with the dative case in verb + preposition idioms. With verbs of fear and protection, it usually means of and from or against respectively.
With this one can decipher the correct case of - hopefully - almost every prepositional verb.