Its great to hear that you are thinking about going vegan! My first suggestion would be to maybe talk to your family and see if you can make some vegan meals throughout the week together? You dont have to eat meat substitutions or eat soy yoghurt or alternative milk or cheese etc to be vegan. Instead eat whole foods, then its less expensive!
It is of course easier to be vegan when you are in charge of buying and making your own food, and when the fridge is filled with only vegan foods, but remember that you have the rest of your life to eat a fully plant based diet. You dont need to rush and move out just because you want to eat vegan. Infact it can be better if you are only just recovered/recovering, if you still live with your family and eat meals with them but maybe implement some vegetarian or vegan meals throughout the week. Inspire your family to eat more plant based meals and to show them how easy and delicous and nutritious it can!
Also remember that vegan isnt just about the way you eat, of course that is a huge thing, but maybe for now you can focus on other aspects such as maybe not buying products tested on animals, and buy items which dont have any animal substances in them such as no wool, no leather, no snakeskin etc ? All of those things play a huge roll!!