Know the purpose
Before you make a strong case to promote vegetarianism among people especially who you think need to hear about the importance of being vegetarian, you need to understand the overall purpose of the effort. It is important that you should gather all the vital information before you can argue or write down any kind of detail about vegetarianism. You should also know the very purpose of consuming vegetables in your daily diet. You can also raise questions about why people should stop eating animals. The awareness of killing animals or any living being is also a big question for people who are not vegetarian. The concept of not eating any kind meat is actually very old since the evolution of Buddhism. However with the passage of time, this concept is now very common in the modern world as well.
Start with small issues
You should start with very small details asking questions in an investigative manner. You can argue that raising animals is a difficult task as you will need a relatively big place while to grow vegetables is comparatively an easy task. This factor alone can make your case heard.
Cost cutting aspect
Meat is usually an expensive commodity while vegetables are not that expensive. You can easily afford them as everyone wants to save money and vegetables certainly give you peace of mind at their availability and affordability.
You can also mention in your case about different kinds of environment issues which are totally related to meat consumption. The causes of global warming and greenhouse effects are related to carbon dioxide emissions from fossil fuels.