This is an interesting question because the Romans don't seem to have viewed religion as a matter of spirituality, like many Americans do today. For the ancient Romans--and many other ancient cultures for that matter--religion was a matter of action and proper practice. A proper relationship with the gods demanded the performance of certain rituals in a certain way. Failure to perform those rituals or failure to perform them in the proper way resulted in the gods being unhappy.
In the Aeneid, some divinities play favorites. Aeneas is the son of Venus, so Venus tries to help her son as much as she can. Juno hates the Trojans and so she tries to do everything she can to harm Aeneas.
We also see divinities like Jupiter serving as the executor of fate or destiny. It appears that Jupiter knows what the Fates have declared and it is up to him to make sure this is carried out:
‘Don’t be afraid, Cytherea, your child’s fate remains unaltered:
You’ll see the city of Lavinium, and the walls I promised,
and you’ll raise great-hearted Aeneas high, to the starry sky:
No thought has changed my mind. (A.S. Kline translation)
Other divinities serve as messengers of the gods, such as Mercury, who tells Aeneas that it is time for him to stop dallying in Carthage with Dido.
Still other divinities function as helpers to other divinities, as in Book 1 when Aeolus, urged and bribed by Juno with marriage to a beautiful nymph, stirs up a storm against Aeneas. Juno also enlists the aid of Allecto to stir Turnus to war. Thanks to an assist from Vulcan, Venus gets some divinely made weaponry for Aeneas in Aeneid 8.
Thus, the gods in the Aeneid play numerous roles. Whether they teach us anything about spirituality, though, I'm not terribly convinced that they do. The Romans didn't view religion as a matter of spirituality. Perhaps the Aeneid and Roman religion teach us what spirituality is not. Modern spirituality seems to be about an effort to have one's "spirit" in tune with the "divine." Perhaps in the Aeneid, one could say that Aeneas' "spirit" comes into tune with the divine when he finally realizes that he needs to leave Carthage and start focusing on his destiny:
Aeneas, stupefied at the vision, was struck dumb,
and his hair rose in terror, and his voice stuck in his throat.
He was eager to be gone, in flight, and leave that sweet land,
shocked by the warning and the divine command. (A.S. Kline translation)