I agree with the suggestions to try cooked versions of sushi like California Roll, cooked shrimp, cooked egg, smoked salmon, etc..
Moving on from that I'd start trying the small raw fish "maki" rolls - like tekka maki (tuna), etc. While these little bite-size morsels do contain raw fish, the combination of ingredients in the roll give you an introduction to the taste & texture without screaming "raw fish".
When/if you're comfortable with different raw fish rolls, try some samples of nigiri sushi from the a la carte menu. This way you usually get just 2 pieces of any selection to try, instead of a whole platter. Start with cooked shrimp & move on to raw tuna & raw salmon (both safe mild bets). And even tho you've never had eel, how will you ever know you likie it unless you try it?? Eel is always cooked for sushi & is DELICIOUS!!! Why wouldn't you want to at least try it?
Being a picky eater is no way to go through life when there is so much delicious food out there to try. And that has nothing to do with just sushi; I mean food in general. I thank my lucky starts that I was raised by parents who made us taste EVERYTHING before proclaiming whether or not we liked it. They never forced us to eat anything after that initial taste, but we did have to taste it first. I shudder to think what my food life would be right now without that education. So many of the people I grew up with were so catered to foodwise as kids that these days you can't give them anything but Spaghetti-O's to make them happy. So so sad.