I don't consider myself especially picky, but I have a few "can't eat" and some "don't like" foods. On the can't eat list are: shellfish and rhubarb (allergic) and pepperoni (ok, technically it won't kill me, but I will be horribly uncomfortable for several hours afterwards). My digestive system apparently thinks tofu is a toxic substance - I actually didn't mind it in the item I tried it in (a frozen meal that included chunks of tofu), but my intestines apparently said: "What the @#$!! is THIS STUFF??? CORE DUMP! CORE DUMP!"
On the won't eat list: melon (I've found that I dislike watermelon, canteloup, and honeydew, so it's easier to just say I don't like melon); spicy food that's "hot" spicy; I don't care for slices or chunks of raw tomato, but will eat salsa (as long as it's mild), and tomato sauces are just fine. I'm not big on green beans, but will eat them occasionally, and I tend to prefer broccoli and cauliflower raw, but will eat them cooked as long as they still have some texture and aren't completely cooked to mush. Oh, and I won't eat jello...unless it's in a jello shot, which comes under different rules.
The rule at my house was I had to try two bites of something. If I didn't like it, I could fill up on other stuff and nothing would be said about it. There were children's menus at some places when I was growing up, but I usually didn't want one. I'd usually just order something like a petite steak or something like that off the regular menu instead. Which delighted my maternal grandfather in particular; he loved a good steak, and loved buying me one.