Good for you for wanting to make a change for your family and please don't listen to anyone who says a plant based diet is unhealthy. Any diet that isn't well thought out can be unhealthy.
Only you can decided what you want to eat. A true vegetarian eats no animal meat (including fish), so it makes me a little nutty when people say they are vegetarians that eat fish...no, they are pescetarians. Anyways.
Vegetarians are meat-free for lots of reasons. I am a lacto-ovo vegetarian, as is my son (for now) which means I eat no meat, but I do eat dairy and eggs. A vegan would eat zero animal products to include dairy, eggs, honey, etc.
If you have no ethical reason to go vegetarian, you could certainly go plant based as much as possible, but include fish on occasion. A sustainably fished, low mercury fish is a very healthy thing to include in your diet. Since I am breastfeeding, I do take fish oil supplements as does my 2 year old.
The number one concern with people going veggie is getting enough protein. You need to determine the amount of protein your family needs based on ages and activity levels, etc and plan accordingly. The good news is that most Americans get way more protein than they need and it is not difficult to get protein.
Milk, all dairy (like yogurt, cheese), eggs, nuts, tofu, beans, nut butters, legumes (lentils), etc all provide protein. Breads, bagels, tortillas, etc all have some protein in them. Breakfast of cereal, yogurt, fruit and some nuts sprinkled on top is a good healthy breakfast.
My motto is everything in moderation and everything organic! :) Try to eat as whole as possible, any processed foods are usually junky and should be reserved for special occasions. There are conflicting views on soy....the bottom line is that if you eat it in moderation and not eat truck loads, you are fine.
There are millions of recipes out there for vegetarians. Check out vegetarian times magazine on line or in the store. They have great, easy recipes. Google "vegetarian" and you will get a ton of responses. Crockpots are a great way to get meals done and have extra for yourself!
If you have reluctant veggie eaters, check out your local farmer's markets, get your kids involved and go buy things you don't recognize. The farmer will tell you what to do with it and it's fun. We made squash blossom tacos the other day...they were yummy! May not make them again but we had fun doing it, using something atypical, and my 2 1/2 year old was thrilled to eat "flowers".
Good luck!
Good for you for making a change