Appa's relationships

Momo
Of all members of the team, Appa is possibly closest to Momo. They are shown communicating several times in the series, such as toward the end of "The Cave of Two Lovers" and "The Swamp". Although in Nightmares and Daydreams the two have an incredibly quick, yet violent, battle (complete with samurai swords and armour), this can be disregarded as a factor in their relationship, since Aang saw this while sleep deprived and hallucinating.

Nyla
In "Bato of the Water Tribe", Appa and Nyla fought each other intensely. When they reunite in "Sozin's Comet, Part 2: The Old Masters", they warm up to each other.

Hawky
Having only been with Hawky in "The Runaway", Appa barely knew Hawky. It was only when everyone left to go into town and Appa was put "in charge" of the animals, that Appa came and settled a dispute between Hawky and Momo.

Aang
As Aang's sky bison, Aang has always used Appa as his primary means of transportation. Aside from this, Aang has promised Appa that he and Appa will always be together, as said when they met. In "Appa's Lost Days", Appa wanted nothing more than to be reunited with Aang. When they finally did in "Lake Laogai", Appa and Aang are both apparently very happy to see each other again, after weeks apart.

Sokka
Although Sokka didn't like Appa when he first saw him, calling him a "fluffy snot monster", and he was skeptical of Appa's flying abilities (until he himself gets him to fly), by the end of Book One, Appa and Sokka are shown to have bonded considerably, and Sokka even uses Appa to impress Yue.

Katara
Katara has always liked, and been sympathetic to, Appa.

Toph
Toph and Appa's relationship was rocky at first because Toph blamed Appa for leading Azula to the group, and because she doesn't like flying. Toph and Appa's relationship improved considerably when Appa returned to the group after he was captured.

Zuko
As an enemy of Team Avatar, Zuko originally was an enemy of Appa. At Ba Sing Se, Zuko freed Appa from the Dai Li. This act was one of the reason Aang and the others accepted him into the group.