Assuming you have been strongly weighing the health benefits of adopting a vegan diet, it's reasonable to expect it will take some time to reach your goal once you begin. While everyone's situation is different, and there is certainly no single correct way (while I'm sure there are some who transformed overnight, the journey took me several months), below are a few suggestions that you may find helpful.
1. Once you have decided it's time to give up meat (pick up a copy of "The China Study" by T. Colin Campbell if you aren't convinced you should), you may want to savor your favorite foods one last time. Use the opportunity to celebrate and make your final taste of that food a memorable occasion. You can even take photos. You may need several days to do with all your favorite foods, so I'd recommend a limit of 10 dishes.
2. Start to try to incorporate vegan products, such as non-dairy milks or soy products (tofu, tempeh, textured soy protein), into your diet. I started ordering cafe lattes prepared with soymilk instead of dairy milk. To be honest, I hated it at first (thought it a waste of perfectly good espresso-and what's worse-cost extra, too!) I also replaced my traditional breakfast yogurt with 5-Grain hot cereal.
3. Order vegan cookbooks that match your particular ethnic tastes (some of my favorites are highlighted on this site), and pick up any ingredients you need to prepare them. Don't worry about the cost, and try out as many recipes as you need to find something you enjoy, or at least feel you could in time.
4. Start eliminating one category of animal food little-by-little. In my case, I cut out red meat and chicken, but kept eating fish and dairy occasionally (sushi and cheese were the hardest things for me to give up). I still had a lot of cheese in the refrigerator, and I thought there were no substitutes for eggs.
5. Eventually, begin to avoid eating animal products whenever practical. However, you don't have to have a heart attack if you learn a dish you're eating has a trace of meat (or dairy product). This is a benefit of becoming a vegan for health, rather than ethical, reasons.
Remember, you may lose weight at first on a vegan diet, but it will return once you discover vegan versions of your favorite foods. Vegan plant-based diets are generally healthier than animal protein-based diets, but it depends on the quality of ingredients and how they are prepared. For example, whole grain breads and pastas (complex carbohydrates) are much healthier choices than non whole-grain products. And oily, overly sweet, or processed foods should also be avoided, whether vegan or not.