The following a few key episodes of the anime that I feel are important to Giovanni. Though they cover only three episodes, I feel that these are the most important to his role in the anime. He appears often in the recent episodes in Meowth's "Boss Fantasies" which are talked about below in the "videos" section. I also provide links to the videos so you can see for yourself. Though these episodes are available to watch online at Pokemon Episode.org, if you are a fan, you should do your best to support the series. Go out and purchase the DVDs and for yourself.
Granted, this is all that the audience sees of Giovanni but it is very significant. It is the first episode that Giovanni is shown or mentioned in the series. This gives the viewer the impression that Team Rocket does have a boss with some sort of nefarious plan. The viewer is then led to wonder what will be become of the heroes. Note also that this episode shows the staple of Giovanni until about 50 episodes later. He is constantly shown in shadow or his face is not in the overall shot. Again, adding that air of mystery around him.
Episode 17 Island of the Giant Pokemon had Giovanni in it for a few seconds at the very end but nonetheless is worth a mention. (I did not include images for the sake that Giovanni is hardly in the episode.) The episode revolves around Ash, Misty, and Brock trying to find their Pokemon who are lost on a strange island where they are being followed by Giant Pokemon! Soon the team reunites after much mayhem and destruction. Giovanni is shown getting a phonecall from someone and then again at the scene of the Park. It is crucial to the plot showing that the island was actually a theme park...owned and operated by Giovanni!
The trio makes it back to Viridian city to find only that Gary has beaten them there and he runs off to challenge the Gym Leader for the Earth Badge. A momentous episode for viewers when Giovanni finally comes out of the shadows to show his true self. Unbeknownst to Gary, Giovanni is the leader of Team Rocket and he does not play fair! This episode also linked with Pokemon the First Movie when Giovanni explains that he wants to try out his "new Pokemon". It turns out to be Mewtwo and crushes Gary and his Pokemon.
Team Rocket later tries to present him with a Togepi, which he rejects, and runs off due to an important call. Ash wins the badge only because Jessie and James are battling. Ash never actually meets Giovanni in this episode, and Gary never really talks about him, only about Mewtwo. The audience never learns why Giovanni had to leave so abruptly but we can conclude simply that it was a plot device for Ash to gain the last badge.
This section is based entirely on Meowth's "Boss Fantasies". I am not sure when it started but it has been an ongoing gag in the Pokemon anime where Meowth has a Fantasy about how Giovanni will react when they send him a particular Pokemon. Most of them are incredibly weird but are enjoyable nonetheless, here are just a few of my choosing. Note: This section was last updated on June 3rd, 2013. Due to copyright conflicts, all of the following shorts were removed from YouTube.
Japanese Only Apparently this clip was cut from the American release because it was just that weird. Meowth envisions Giovanni covered in honey and running through the wilderness only to be attracted by hundreds of bug Pokemon, which then cover him and lick(?) the honey off of him.
The Poster of this video states that this is the Earliest of Meowth's "Boss Fantasies" and comes from the Johto arc (3rd season) of the series. Jessie, James, and Meowth are enticed by a Mantine and aim to get it for the Boss so he can enjoy swimming in his pool.
Jessie, James, and Meowth come across a slew of Prehistoric Pokemon. They originally want to sell them until Meowth tells them to give the Pokemon to the Boss. He says they will be highly rewarded when he comes to the island and the primitive Pokemon awaken his "Primitive side". He then impersonates Tarzan.
Team Rocket plots to steal a Psyduck but Meowth suggests to give it to the Boss instead. James's commentary on Meowth's senseless babble is the most amusing part of this clip, aside from Giovanni building a dog house.
Deoxys appears and offers to take Giovanni into space, because he apparently always wanted to be an astronaut.
After seeing how strong and tough Nurse Joy's Chansey is, Meowth suggests that they capture it to give it to the Boss so he can have a good fight. Apparently he likes to get beat up by squishy Pink Pokemon?