First of all, are you sure it's the meat that's the problem?  A lot of people who think that meat is making them sick are actually being made sick by additives, food substitutes and other things that are cooked in or with the meat.  For example, the foods that you mentioned are often fried or "grilled" on a flat grill, where they stew in their own oils as well as any oils added to keep them from sticking, and (especially at burger joints, where there's pressure to be "non-transfat") some modern oils used for frying are known to give people digestive problems.  I can't eat anything with Olean/olestra ( en.wikipedia.org/wiki/Olestra ) for example.  I also have strong reactions to Nutrasweet (aspartame) and Splenda (sucralose).  This severely inhibits what I'm able to eat of processed foods these days, because many, many products are made with these three additives.  (The next time you're in the water/drink aisle of a store, for example, check the labels of products such as Propel Fitness Water, generally all diet sodas, and other items marketed as "healthier" alternatives to traditional sweets and snacks.)

Generally speaking, it IS better to eat organic, raw and with a larger proportion of fruits and vegetables anyway.  Those of us who are @$$-broke can't always afford that luxury.  (Compare the price of organic orange juice to a similar size of soda, for example; and if you want a REAL sticker shock, compare the per-ounce price.)  Thus, one of the things to take into consideration is that you will most likely need a larger budget for food, at least until you figure out what you like, what you don't like, and how to maximize your meals with a minimum amount of waste.

Often, people who say they "don't like fruits and vegetables" have simply not had them prepared properly.  A common example is spinach - people swear up and down that they hate spinach, yet most have only ever had it cooked.  I don't like cooked spinach, and it's a waste of a lot of the beneficial aspects of the food.  (Typically, the more you cook a food, the more its nutrients, minerals, and food body are broken down and destroyed before you ever have a chance to consume them.)  Many people can't tell the difference between raw spinach and the green part of lettuce, especially if it's served in salad, on a sandwich, or otherwise presented as lettuce.  It does have a stronger and more sharp flavor than lettuce, but for those with minimal exposure to either, the difference isn't always as noticeable. The same goes for other vegetables, herbs and fruits that people say they hate - RAW, RAW, RAW.  Try it raw before you decide you hate it; cooked vegetables and fruits (of any degree - all the way from slightly steamed to stewed to fried) have a much different consistency (which matters to people who are texture-sensitive) and often different flavor than their raw counterparts.

Diversify your diet of fruits and vegetables.  "Eat your greens" is something that a nagging mother would say, and you do need them; they're often an important source of iron and other minerals.  However, a good rule of thumb is to "taste the rainbow" - try to have some fruits or vegetables of every color during the day.  Reds (apples, strawberries, cherries), Oranges (oranges, carrots), Yellows (bananas, pineapple), Greens (lettuce, spinach, broccoli), Blues/Purples (blueberries, eggplant, grapes, raisins) are all available fairly commonly - you just have to train yourself to notice the colors of the food you're buying.  (This will also eventually lead to you being able to better choose fruits and vegetables that are in healthy condition and ripe instead of being over/underipe or in lesser health.) 

You can also include fruit and vegetable juice in your diet, but be sure to read the label and make sure you're getting JUICE and not a load of chemicals with some fruit/vegetable flavoring in it.  Because of my sensitivity to food substitutes, I can't drink V8 Splash - along with the fruit juices, it also has sucralose in it to make it sweeter.  Other beverages will use high fructose corn syrup to sweeten the drink.  Organic, well-made fruit and vegetable juices DO NOT need additional sweeteners - the Western world has just developed a severe addiction to the taste of sugar since the introduction of processed sugars to our diet.

I did a little googling and found a photo that shows just how easy it is to find a strong variety in fresh fruits and vegetables for a healthy diet:

(I've had all of the foods in that image, btw, all are good when prepared properly, and most have many, many options for meals and preparation.)