I was in the same situation as you when I started. I had just watched Naruto and Bleach and really enjoyed both, I heard that One Piece was even more popular and better than Naruto and Bleach so I tried the anime. First time I stopped at episode 2 and on my second try I couldn't get passed episode 5.
This was untill started reading manga instead of watching anime. I wanted to at least get to Arlong Park, since that is where most of the fans say it gets good and after still not liking it at first, I got hooked at Baratie and it has been my favorite manga ever since.
The cartoony style gets less and the story, characters and world building are amazing. It can grow on you, but it needs some time for most people.