I am a vegetarian by choice but I would like to give meat to my four-year-old daughter. Should I start with chicken?

Yes chicken would be a good choice. Since your daughter is already four, any meat would do. Chicken and turkey won't have that much odour and might be easier for your daughter to accept at the beginning. For Beef, fish, and pork, I'd suggest to mix it with other odours, for example spaghetti bolognese or fried rice for shrimps etc.