I am working on getting more definition in my abs, so I was cutting sugar out of my diet to help shed those last few pounds. I haven't tried cutting out sugar before and had a few questions. First, I wanted to know about whole fruit. I assume I can still have sugar that is in whole fruit, but are there any fruits I should avoid? second, am I right to assume that small amounts of refined, or processed sugar(3 to 6 grams) will probably be used up by my body and not be stored as fat.

You chose this adventure, so you're going to have to write the rulebook, too. There's no "no sugar" diet with rules to be followed.

If you're in a caloric deficit, all of the calories you consume will be used by your body and not stored as fat. If you're in a surplus, your body will use some of the extra energy to put on fat stores. It matters not whether that extra energy came from refined white sugar, fat, protein, or all natural organic planted-tended-and-harvested-by-virgins superfoods.