In Who Gets to Keep Togepi?, Ash, Misty, Brock, and Meowth fought over the Egg and competed in a battle tournament to determine who would be the eventual owner. When the Egg hatched into Togepi, the first thing it saw was Misty, and it subsequently chose her as its Trainer, thinking that she was its mother. This was in spite of the fact that Ash officially won the tournament.
In The Battle of the Badge, Ash was startled by Togepi and accidentally sent it flying away, where it eventually ended up in the hands of Team Rocket. They duly presented it to Giovanni who was unimpressed as Togepi couldn't actually do anything remarkable at the time. Togepi managed to slip away and soon reunited with Misty.
As a Togepi, crying
Although Togepi appeared harmless, it sometimes used Metronome to get out of dangerous situations. The first time this happened was in Pikachu Re-Volts. Butch and Cassidy's Drowzee was using Metronome to attack Ash and his friends. Misty's Togepi began using it, causing an explosion that sent Butch and Cassidy flying. It happened again in In the Pink. A Rhyhorn was about to charge the group when Togepi used Metronome to use Teleport. It also used the move later in the episode to stop a Nidoking's Hyper Beam by using Barrier.
Despite the fact that Misty knew Togepi could battle, she only used it one time to battle Ash. In The Totodile Duel, Ash was using his Pikachu to battle Misty for ownership of a Totodile. Misty used Togepi, because she knew Pikachu would never attack it, as Pikachu would often babysit it. Using Charm, Togepi ended up defeating Pikachu.
In Houndoom's Special Delivery, Togepi was separated from Misty and the rest of the group for some time, after it used Metronome to teleport itself away from James's Weezing. A Houndoom rescued it from a tree and the two became good friends, sharing a sad goodbye when it was time for Togepi to leave.
Togepi later evolved into a Togetic in A Togepi Mirage! in order to protect the other Togepi. When Togetic was injured by Colonel Hansen, Misty promptly sent out her Gyarados and defeated Hansen's Shedinja with a Flamethrower attack. After seeing the damage Hansen had done trying to reach the Mirage Kingdom, Misty respected her Togetic's decision to stay behind and released it, so it could protect the Kingdom, the Togepi Paradise, and the Togepi who lived there.
As a Togepi, it was close to Misty ever since its hatching, and they quickly developed a relationship akin to that between a mother and her child. However, despite its adorable appearance, Togepi boasted an incredible amount of power that showed itself on occasion.
As a Togepi, it was particularly curious about the world around itself. Because of this, it was prone to walking off and getting itself and Pikachu who would always pursue it into dangerous situations, which usually resulted in Pikachu getting hurt while Togepi walked away unharmed. It was very friendly towards other Pokémon, particularly ones that were small, young, and wild, such as a Sentret in Tanks a Lot! and, much to Misty's dislike, a Caterpie in The Bug Stops Here. It also made friends with Pokémon who had a fearsome reputation, such as Tracey's Scyther. Again, it would be unaware of the potential dangers like Voltorb in The Underground Round Up.
Misty is shown with a Togepi in The Pokémon Banana League (Part 2). She temporarily gives Red her Togepi when he was battling his rival Green, as Red couldn't handle him on his own. Togepi was returned to the rightful owner after the battle was finished.
Togepi was also the first baby Pokémon owned by a main character.
Togetic is the first Pokémon to be seen hatching from an Egg in the anime.
Togetic is referred to as a male in the Brazilian Portuguese dub of several episodes. However, this is unlikely to be an indication of its actual gender, since the neutral pronoun in Portuguese language ("isso" = it) is not used to directly refer to someone or something, instead "ele" (he) or "ela" (she) have to be used even if the gender of the subject is unknown.
For more information on this Pokémon's species, see Togepi and Togetic.