My family are annoyed because I'm vegetarian?

If you don't like the restaurant they have chosen, my advice is to let them go out and have a good time, while you eat whatever you want at home. Or wherever.

Never EVER heard of a restaurant that had NO or few vegetarian options.
they all have potato of some kind, even a steak joint has baked potato and salads available.
You don't have to "get across" anything to them. don't eat meat, don't make a big deal of it and choose those veggie options that ARE available. A quick call to any restaurant will tell you in a hurry what is or isn't available to a vegetarian.
No stress.
don't argue.

One assumes that you're old enough to stay home alone.
Tell your mother to stop shouting, because you're not going.
Or go, and eat your salad without complaining.

get a salad with no dressing, its simple! And ive yet to hear one resteraunt that dosent serve salads,
or stay home or pack a meal OR eat before, i do one of those things, depends on the time (like if they went to an icecream shop (and im vegan) i would stay home, bt if they were going to a fancy resturaunt down town, i would eat before) and for the bully part, i would just look down at my plate and/ or say "its my decision, and i dont shove a piece of tofu in your face annoying you about it"

Don't get angry, no matter how hard it is. This will show that you are irresponsible. Explain calmly that this is your decision. Say that you understand her frustration, but it is your choice. Maybe suggest going to somewhere with a veggie resturaunt, research good places. I really hope this help, good luck and what ever you do, don't get angry.

She shouldn't be screaming at you like that. But there are veggie options at restaurants, especially now a days, with restaurants being more respectful of vegetarians. If nothing else, maybe you can have a nice salad. Most restaurants have salads that are pretty robust and would make a good meal. Even fast food restaurants.

You only need one option.

Many restaurants are willing to modify a dish for vegetarians. Say it's a pasta dish with chicken. Ask for no chicken, and hopefully they will take some money off. If they won't, ask for the chicken on another plate and give it to someone else while you eat the pasta. There are also many salads (and while I have encountered house salads with bacon bits) that have no meat or have an option for no meat. It doesn't hurt to ask for no meat, or to ask for it to be separate.
As for talking to your family, just explain that it's a personal choice and they don't have to agree with it, as you don't quite agree with theirs to eat meat (not that you're trying to change their minds). You have your reasons, and don't keep defending your choices if you don't feel its worthwhile.

take a packed lunch

Every place has a plate of spaghetti with tomato sauce, doesn't it? (That's what I usually ask for).

http://veganlifenow.blogspot.com/

When go to non-vegetarian restaurant and if they don't provide vegetarian dishes.
Try order mee (noodle) and ask them not to put meat/egg into the mee.
Most restaurant (even non-vegetarian restaurant) does provide vegetable dishes, ask the waiters that you don't want meat/egg added to the vegetable dishes.
I some time did that when go to non-vegetarian restaurant because couldn't find any vegetarian restaurant nearby.

There is always veg option so pick a veg option unless your one of them new vegetarians that have no clue what vegetables are!!

Too many veg/vegans being veg/vegans and have NO clue about their diet nor what they talk about. It's embarrabsing

If you're going out to a restaurant, they should be making the food from scratch. This means you can ask for a meat dish without the meat. For example, if there's chicken salad, ask for the salad without the chicken.

You're perfectly entitled to your opinion, some people are grossed out by the concept of eating a corpse and it's hardly surprising or strange. Such tastes aren't a choice and you shouldn't be forced into it. She can yel all she wants, but it's not going to change anything.

Why don't you go with them where she wants to go and just not eat. That's what my mother would have done.