I think that Pokemon does do a lot of things right, and that there isn't much to fix with it.
First of all, the game balances complexity with simpleness really well. Any 9 year old can pick up the game and start playing it. The type match ups are easy to figure out and work with in order to enjoy the game. There is also a deeper part to the battle system as well. The stats menu, which I ignored as a kid playing the games, add a whole new strategic plan when battling. There is also a mathematical formula on how the moves affect the opponent. As I already said, the game keeps it simple but has a deeper element to it that, when harvested, can make the battling much more satisfying.
I think we need to remember too that Pokemon is marketed mostly to children. If you were an adolescent in the late 90s you can remember how exiting Pokemon was. If we want to play them now and enjoy them just the same we have to just get used to the game being marketed towards a younger audience. We don't need Pokemon dying, we don't need Team Rocket with guns. To me, that kind of realism would ruin the game. There is enough violence in the world that we don't need to corrupt Pokemon. Leave it as it is. If you want to play a realistic rpg play an Elder Scrolls game. The game does do a lot to make it fun and deep enough (gameplay-wise) for older audiences.
One of my favorite things about the game is the relationships you can have with your Pokemon. You can do things to make them love you, which feels good, in a way, after you trained them so hard. Giving them natures (which affected how your Pokemon performed in battle) in R/S/E and, in my opinion, even greater, the characteristics in D/P/P was brilliant. That gave Pokemon more than just a sprite on a screen. Abilities, breeding, genders, they were all great ideas as well. All of them added new strategies for battling and raising and were just plain fun to see.
There are some problems I would like fixed though. The biggest problem with the series, as many of you stated earlier, is that there is no innovative storyline. The same story has been recycled over and over throughout the series. I have been playing Final Fantasy V lately and have found the story to flow through very nicely. Of course, it is a corny "unlikely hero/save the four elemental crystals to save the world/guy with amnesia/finding out that the guy in your party is actually a girl and is the long-lost sister of the princess who travels with you to revenge her father/etc."(I really didn't ruin anything. You find out early in the game and it isn't that hard to guess) story, but it works. Now I'm not saying that I want the story to be completely corny, but having a deeper storyline that isn't the same generic Pokemon story line that we all have played over the past twelve years would be great.
Another thing that I have always wanted to see in a Pokemon game is a chance to take different paths to get to the end of the story. If I may, the storyline plays like a rail-rpg until you get to the end, where you can jump off of the rail and go back if you'd like. Having side-quests that allowed you to do this would be great.
Now, about character creation, there is a part of me that says yes and a part of me that says no. Yes, it would be fun to create your character. That is one of my favorite things to do in a video game. I could spend all day creating characters on The Sims 2 or Oblivion and not get bored, but Pokemon, to me, doesn't seem like it is a character-creating game. My intuition tells me that it would take away from the Pokemon charm. Maybe choosing different avatars would be great, but besides that I don't know.
Pokemon-wise, I disagree about disowning all Pokemon past the first generation. There are some great Pokemon that have come out in the second and third generation that I love. (I have tried to not look at all of the fourth generation Pokemon. I want what I find in the grass to be a surprise when I finally get the game.) There are some Pokemon that aren't that all exciting but one has to look at the diversity and creativity of the pokemon in the pokemon world and admire them for those creations. There are bound to be some questionable Pokemon when you have created so many.
There are some more small things with the game that I would like to see fixed.
I personally found the contests in Pokemon Ruby/Sapphire/Emerald to be a lot of fun. I only wish that they actually accomplished something in the game. Fixing that would be great.
Oh, I almost forgot this one. The variety of Pokemon that trainers you fight have needs to be greater. It gets old after a while battling the same Breeders or the same Hikers with the same Pokemon in tow.
That's all I can think of for now. I'll try to come up with some more ideas.