See how Patamon is leaning against Gatomon like that? It's so cute...]
In Episode 7 Patamon digivolves to Angemon for the first time in the season. He could have just armor digivolved to Pegasusmon. Why did he choose Angemon now of all times? Probably to impress Gatomon. But unfortunately this didn't work...see how mad Patamon was when he regressed? "Man, those control spires really ruin my day."
In the famous Episode 13, they are seen in a tree together, probably for light conversation, and Patamon says school looks like fun. THEN WHY IS HE IN THE TREE? (And in the Japanese version Patamon was the one who pointed out that Hikari didn't look so good, not Tailmon. Patamon knew that Tailmon already knew that Hikari looked bad. Ah, he just wanted to talk to her.)
In Episode 22, Gatomon is dancing with Patamon, not DemiVeemon, and I don't think it had anything to do with Veemon being a bad dancer.
And, of course, they're constantly seen with each other througout the series.
But that's where it ends (unfortunately).
An interesting thing to note is that their Champion-to-Ultimate digivolutions are similar. First, the common Crest-and-Digimon-spinning is shown. Then, the Digimon is silhouetted in light, which changes form into the new form. The details are added (i.e. Angewomon's ribbon, MagnaAngemon's little arm-thingies), and the eyes are briefly seen before they are covered by the helmet and put back into the final pose.