What do you mean by "fight inflammation?" If you have an inflammatory disease like severe arthritis, your doctor should refer you to a registered dietitian with a subspecialty in your condition.
If you just mean "I read some stuff that says inflammation is terrible and we're all gonna die from it," don't worry about it. There are a lot of people out there trying to scare us into buying their books by going on Dr. Oz or paying to have their books mentioned in content-farm "articles" that get published in struggling magazines. If you're reading about inflammation in Woman's World or Prevention or a magazine like that, somebody is making money by getting you to think it's a huge problem while not explaining what it is. If your medical doctor hasn't run tests and given you a diagnosis of inflammatory disease, then inflammation isn't high on the list of things you need to worry about. A generally healthy diet like you've read about to help prevent heart disease, obesity, etc will also help reduce inflammation; inflammation is just one specific part of a lot of diseases.
That said, going sugar free isn't a bad idea. Nobody needs refined or added sugar, and it's not that hard to avoid it. The meal plans at Spark People don't have a lot of refined sugar. When they do, it's in something like flavored yogurt or spaghetti sauce where you have a choice of brands and you can just make sure you choose a brand without the added sugar. If you want a pre-written meal plan, just use the regular one, and if it calls for something that has sugar, you can substitute something that doesn't.
And what "people" mean when they say no sugar depends on who the "people" are. Most people mean "no added sugar" or "no refined sugar." Any plant food or dairy product is going to have natural sugar, so you can't go literally "no sugar" and still be healthy; you would be limited to meat. So the way to know whether a food has added sugar is by looking at the ingredient list, NOT the nutrition facts label. The nutrition facts usually lump together all sugars, natural and added. In the ingredients label, look for the words sugar, syrup, or cane. Depending on how serious you want to be, you could also include molasses and honey. Nutritionally, they're no different from corn syrup, but in practical terms, they don't show up in many products and when they do, they're in small amounts.
Also, look at where sugar comes in the list of ingredients. Ingredients have to be listed in order by the amount that's in the food, starting with the biggest and ending with the smallest. So, for example, if you have a loaf of bread and it has sugar or corn syrup listed *after* yeast, that means there's less than 1/4 oz in the whole loaf. A slice would have less than half a gram. You can decide for yourself whether that really matters enough to justify buying a sugar-free version of that product.
The main benefit of saying "no added sugar" is that it helps you choose foods that are healthier in other ways and gives you an extra motivation/excuse not to eat things you know you shouldn't eat. If someone offers you a donut, it's a lot easier to say (and think) "No thanks; I'm on a no-sugar plan" than it is to say, "No thanks, my body doesn't need that." Any time you eliminate an ingredient or a food, you're likely to lose weight for that reason. Refined sugar is a good one to use, because it's not in very many healthy foods, and because other people will understand what you mean and won't give you a hard time about it.