I've been a vegetarian for about 15 yrs now. Never had any health problem related to food (other than loving to eat a little too much...LOL). My cholesterol is excellent, even for my age & weight. I love being veg and I don't miss meat at all. I'd like to be vegan but I'm still sorting out how to make it work.
I think you'll find that there is a resistance to going veg because our culture puts such emphasis on meat. I bet they say "where do you get your protein from?"
The whole protein thing is grossly exaggerated. You can get the proteins you need as long as you're eating a balanced diet (especially if you're doing eggs or dairy). Vegetarians has been advised to eat complimentary foods to get their complete protein needs, but recently some people are saying we don't even have to eat the complimentary foods at the same meal. It's important to eat a balanced diet (which is true even if you do eat meat).
The main thing you may lack is B12, but you can get it from eggs and milk if you aren't totally vegan. If you're vegan, check out nutritional yeast. Some people worry about iron, but if you're eating right you're getting iron from green leafy veggies and other sources.
A good multi-vitamin may be a good idea, especially in the beginning as you're learning more about food & adjusting your diet. I've been doing this 15 yrs and I'm still learning new things about nutrition & food every day.
I agree with the others who said be careful not to be a junk-food vegetarian. It really is easy to slip into that habit, especially when so many restaurants & frozen dinners 'contaminate' many entrees with meat, meat broth, fish paste, etc. They tend to think vegetarians should be eating a big cheese pizza and some iceburg lettuce. You may be surprised how many foods have animal in them. Even McDonald's french fries are coated in animal products!
Best of luck to you!