you'll learn the ingredients to avoid, just read labels, which you are probably pretty used to from being vegetarian for so long. there are lots of resources or links on this website for hidden ingredients if you look around.
I know it can be frustrating to live with a family that isn't very supportive of your decision. It seems so simple to me and you, but they don't understand/agree and you have to respect that. In a way, it's pretty unfair to expect them to change their diet for you, as it can be quite hard to find things without dairy/eggs nowadays. I'm not saying you can't go vegan, you can and should, but it has to be through your effort not theirs.
Start by going with them to the grocery store, and pick out more produce and things that you can eat. The more simple and less processed things you buy the easier it is to avoid animal products. Buy lots of veggies/fruit. you can buy a tub of quick oats and make a good breakfast out of that by adding applesauce/cinnamon, or other fruit. Try to plan a few meals that you can all eat, even something they could add meat to if theyd like. it's pretty easy to make an awesome vegan burrito - black beans, tomatoes, avocado, onions, peppers, corn, lettuce/spinach, olives, w/e u want. they can add meat if they want, sometimes my family doesn't even bother.
educate yourself about nutrition in general and about vegan nutrition, so you can stand up for yourself when someone tells you there's not enough protein or calcium or some junk like that
hope that helps. go for it, not trying to discourage you but it's definitely attainable if you put in some effort. you can message me any time if you have a question.