Of the three types of love you mention, agape and phileo are found in the Bible. The third one, eros, is not in the Bible.
There's one often-cited passage that nicely displays both agape (ἀγαπᾷς) and phileo (φιλῶ, φιλεῖς). I'll show it in both English and Greek, highlighting the words meaning love.
John 21:15-17 (ESV)
15 When they had finished breakfast, Jesus said to Simon Peter, "Simon, son of John, do you love me more than these?" He said to him, "Yes, Lord; you know that I love you." He said to him, "Feed my lambs." 16 He said to him a second time, "Simon, son of John, do you love me?" He said to him, "Yes, Lord; you know that I love you." He said to him, "Tend my sheep." 17 He said to him the third time, "Simon, son of John, do you love me?" Peter was grieved because he said to him the third time, "Do you love me?" and he said to him, "Lord, you know everything; you know that I love you." Jesus said to him, "Feed my sheep.
ΚΑΤΑ ΙΩΑΝΝΗΝ 21:15-17 (SBLGNT)
15 Ὅτε οὖν ἠρίστησαν λέγει τῷ Σίμωνι Πέτρῳ ὁ Ἰησοῦς· Σίμων Ἰωάννου, ἀγαπᾷς με πλέον τούτων; λέγει αὐτῷ· Ναί, κύριε, σὺ οἶδας ὅτι φιλῶ σε. λέγει αὐτῷ· Βόσκε τὰ ἀρνία μου. 16 λέγει αὐτῷ πάλιν δεύτερον· Σίμων Ἰωάννου, ἀγαπᾷς με; λέγει αὐτῷ· Ναί, κύριε, σὺ οἶδας ὅτι φιλῶ σε. λέγει αὐτῷ· Ποίμαινε τὰ πρόβατά μου. 17 λέγει αὐτῷ τὸ τρίτον· Σίμων Ἰωάννου, φιλεῖς με; ἐλυπήθη ὁ Πέτρος ὅτι εἶπεν αὐτῷ τὸ τρίτον· Φιλεῖς με; καὶ εἶπεν αὐτῷ· Κύριε, πάντα σὺ οἶδας, σὺ γινώσκεις ὅτι φιλῶ σε. λέγει αὐτῷ ὁ Ἰησοῦς· Βόσκε τὰ πρόβατά μου.
Note how Jesus uses agape the first two times, and phileo the third time (after which Peter is grieved). Peter uses phileo each and every time.
Now I'll admit that I'm a hacker, not a Greek scholar. But it indeed does seem like these words could have separate meanings. There's probably quite a lot of research done (so room for better answers), but the quick and simple way to get at the differences is just to trust Strong's Concordance and see what it has to say:
agape: charity, love.
From agapao; love, i.e. Affection or benevolence; specially (plural) a love-feast -- (feast of) charity(-ably), dear, love.
phileo: to love
From philos; to be a friend to (fond of (an individual or an object)), i.e. Have affection for (denoting personal attachment, as a matter of sentiment or feeling; while agapao is wider, embracing especially the judgment and the deliberate assent of the will as a matter of principle, duty and propriety: the two thus stand related very much as ethelo and boulomai, or as thumos and nous respectively; the former being chiefly of the heart and the latter of the head); specially, to kiss (as a mark of tenderness) -- kiss, love.
The words agape and phileo are indeed distinct.