Hi and welcome! I am not a vegetarian but the more I read, the more grossed out I become. I am slowly but surely eating a more plant-based diet. Like you, I am still new to all this and need more information. It has been difficult for me to change b/c it's a new way of approaching meal planning and I need help!
I have 1 DH, 1 13 y.o. stepson visiting frequently, a 5 y.o. picky son and a 2 y.o. son. The 5 y.o. will be my biggest challenge. You might try having your 9 y.o help with meal preparation. He might be more inspired to eat something he cooked himself. The twins could help out too! Just keep putting healthy food out for them and hope that eventually they will try it and like it. Check out
www.veganlunchbox.com for more ideas.
I suggest starting at your local library and reading as much as you can. Vegetarian is a broad term. Lacto-ovo vegetarians eat eggs and dairy products. Vegans do not. Not just the ethics but also for health reasons it's advisable to be a vegetarian or a vegan. There will be cookbooks for you to check out and try for free before you spend $ on buying.
I also suggest visiting your local heath food store. Get someone to watch your children so you'll have plenty of time to browse without interruption! There are lots of products that are substitutes for meat, including "chicken" tenders and "burgers" that your oldest might like. I personally don't like the flavor of the substitutes but he might not mind.
Edamame is young soy beans. I buy them already shelled from the freezer section. I love them! They're great to have around as a snack. If your kids like fresh green peas, they'll probably like edamame. Try hummus too- my 2 y.o. loves dunking his Triscuits in it.
Good luck!