Actually, both of those sentences are incorrect. The correct particle to use would be で, which properly expresses "I will go by car". 車を行きます does indeed mean that the car goes, but the 私は at the beginning puts the focus on the you and the actions which are doing. As a result, the sentence becomes incorrect. (As for me, the car goes). が is also incorrect if you are trying to express that you are going by car. が states the grammatical subject of the sentence, and を links the verb to its direct object. They are not really interchangeable. For example, 「ギターが弾ける」means "I can play guitar". You use が in this case because you are not actually directly playing guitar. You are simply able to do it. It would not make sense to use を because there is no action being directly done.

If the sentence is I'm going TO(towards) my car, that would be 車に向かいます If the sentence is I'm going by car, that would be 車で向かいます or 車で行きます