Oh boy, let me go back to my childhood...
For starters, the Game Boy Tower. I had a handheld, yes, but there was something fun about popping in my Yellow version and just playing it on the TV. Also got more convenient when one of us got our Game Boy Color broken *coughmecough* so we used Pokémon Stadium.
The Stadium was a lot of fun. Prime Cup is my favorite if only because the level is consistent at level 100. Pika Cup has to be my least favorite because of Lass. I hate her so much, even more-so than the Fisherman with the Dragonair.
Vs. Mewtwo we only ever did maybe a couple of times. He's freaking hard, I think we only beat him once and got him added to the roster. I think... unless that only happens after you beat him in Round 2. (Unless you could actually beat him, ha.)
The Gym Leader Castle was probably the most enjoyable challenge as a kid. There was just something really cool about actually going through and challenging the Gym Leaders outside of the main game. Unfortunately, though, once you reach the Elite Four and you beat them, you go through immediately to challenge the champion. I don't think you get the choice to change your team or anything. I could be wrong because one: I don't have the game on me right now, and two: I only ever got to the champion maybe twice. I said it was an enjoyable challenge, I didn't say it was easy for me.
Free Battle was cool, especially since I have a big family, so it was a great way to challenge my brothers one-on-one.
Victory Palace was a nice way to check out the 3D models of the Pokémon as well as hear a great piece of music.
Kids Club will always be a personal favorite, though. Again, because I come from a big family, my brothers and I always play together, so if we weren't battling each other, we competed against each other and acted like we would during Mario Party. It was lots of fun. I tend to be the better of the three (well, now four even though the younger two may play every now and then) of us, but there are some of the mini-games I just can never get down such as the Drowzee one.
Ah, good times.