As has already been pointed out, the progression is
Jesus ἀγαπᾷς ἀγαπάω Verb Second Present Active Indicative Singular
Peter φιλῶ φιλέω Verb First Present Active Indicative Singular
Jesus ἀγαπᾷς ἀγαπάω Verb Second Present Active Indicative Singular
Peter φιλῶ φιλέω Verb First Present Active Indicative Singular
Jesus φιλεῖς φιλέω Verb Second Present Active Indicative Singular
Narr. Φιλεῖς φιλέω Verb Second Present Active Indicative Singular
Peter φιλῶ φιλέω Verb First Present Active Indicative Singular
The crux of this passage is the use of different root words in this exchange. Much stock is placed on the differentiation between the two lemmata with the primary distinction being something along the lines of ἀγαπάω as a standard of God's love and/or the way that the people of God should love each other. When this distinction is made, ἀγαπάω is usually held over against φιλέω which, apparently, is a baser skill or imperative.
Grammatically, there's nothing significant to note about the tenses, voices, or moods in this passage. Present, Active, Indicative may carry some imperatival force in this context and Aktionsart would only be useful to the degree that it tells us that the author is describing an event as it occurred in real time.
Verbal aspect theory would look at this and note that this conversation is the focus of the narrative, but that in the foreground is Peter's first response that Jesus knows (οἶδα) that Peter loves (φιλέω) him. What's significant is that the specific morph of οἶδα that is used is οἶδας (οἶδας οἶδα Verb Second Perfect Active Indicative Singular). Verbal aspect would note that the perfect would emphasize Peter's acknowledgement that he absolutely knew that Jesus knew the answer to the question. Aktionsart would say that Peter knew that Jesus has always known and currently knows that Peter loves him.
Otherwise, the verbs themselves are not heavily marked, especially in light of the perfect that occurs within the passage.
As for the words themselves, looking at the occurrences in the SBLGNT, here's what we have:
+--------+-----+
| ἀγαπάω | 143 |
| φιλέω | 25 |
+--------+-----+
and here are the occurrences by book:
ἀγαπάω
+--------+-----------------+----+
| ἀγαπάω | 1 Corinthians | 2 |
| ἀγαπάω | 1 John | 28 |
| ἀγαπάω | 1 Peter | 4 |
| ἀγαπάω | 1 Thessalonians | 2 |
| ἀγαπάω | 2 Corinthians | 4 |
| ἀγαπάω | 2 John | 2 |
| ἀγαπάω | 2 Peter | 1 |
| ἀγαπάω | 2 Thessalonians | 2 |
| ἀγαπάω | 2 Timothy | 2 |
| ἀγαπάω | 3 John | 1 |
| ἀγαπάω | Colossians | 2 |
| ἀγαπάω | Ephesians | 10 |
| ἀγαπάω | Galatians | 2 |
| ἀγαπάω | Hebrews | 2 |
| ἀγαπάω | James | 3 |
| ἀγαπάω | John | 37 |
| ἀγαπάω | Jude | 1 |
| ἀγαπάω | Luke | 13 |
| ἀγαπάω | Mark | 5 |
| ἀγαπάω | Matthew | 8 |
| ἀγαπάω | Revelation | 4 |
| ἀγαπάω | Romans | 8 |
+--------+-----------------+----+
φιλέω
+-------+---------------+----+
| φιλέω | 1 Corinthians | 1 |
| φιλέω | John | 13 |
| φιλέω | Luke | 2 |
| φιλέω | Mark | 1 |
| φιλέω | Matthew | 5 |
| φιλέω | Revelation | 2 |
| φιλέω | Titus | 1 |
+-------+---------------+----+
A brief glance at these numbers shows that John is more invested in using words in the semantic domain: Attitudes and Emotions, sub-domain: Love, Affection, Compassion (Louw-Nida).
A quick search on the Perseus Hopper reveals that, outside of Flavius Josephus, ancient authors had little interest in ἀγαπάω, but φιλέω was much more prevalently used in comparable literature.
Does this mean that this was a divinely-appropriated word that was withheld from human use until the time of Jesus? Probably not. In fact, this rarity increases the probability that ἀγαπάω was a relatively unused word in contemporary literature and was appropriated by Christians to convey their emerging sense of affection.
To force such a distinction between ἀγαπάω and φιλέω is disingenuous. There are places where this distinction is inappropriate (2 Sam. 13:4 Amnon ἀγαπῶ Tamar, John 5:20 The Father φιλει The Son, 2 Tim. 4:10 Demas αγαπησας the present world, 1 John 2:15 Do not αγαπατε the world, John 3:19 the men ἠγάπησαν the darkness).
An alternative solution is that in this instance, the author was attempting to use variety within the text1, assigning words to operators in order to maintain a clear narrative that comes to a satisfying conclusion.
In conclusion, while the general tendency of NT authors is to use ἀγαπάω to convey a higher sense of meaningful love, we need to make sure that we're reading the text responsibly instead of forcing an assumed meaning of a word into the context of the story.
1Craig Blomberg, Jesus and the Gospels 2nd ed., (Nashville: B&H Publishing, 2009), 418.