Congratulations, and I'm glad to see parents being responsible and caring for their children's health.

First, one note, vegetarians do not eat fish, but piscavores do. They are sometimes lumped in as vegetarians, but vegetarians eat no animals at all. So I'll try to answer your question without assuming whether you intend to eat fish or not.*

Eating out as a vegetarian can sometimes be a bit difficult, but you can usually find one or two meals on the menu that work. You increase your chances if you pick places like Olive Garden or Case Ole rather than steakhouses or BBQ places. For fast food, places like Taco Bell are easier than McDonald's for vegetarian meals. Basically, avoid places that heavily push meat and you'll be fine with the menu.

Eating out as a piscavore is extremely easy, as almost every restaurant has a few meals that are either vegetarian or only have fish.

Since you are a on a budget though, it would probably be best to eat out rarely, or never. Fake meat products are also something to buy sparingly, since they can be costly. The good news though, is that fresh meat is also costly, so taking meat out of your diet should make the grocery bill lighter. Beans and rice are cheap, as are noodles and pasta sauce. Canned fruits and veggies are cheaper than fresh and frozen, but I'd suggest frozen since they are healthier than canned (and sometimes healthier than fresh), and cheaper and last longer than fresh. Tofu ranges greatly in price...it will be cheap in Asian markets, mid-range at the Walmart, and expensive in health stores. It's great for Thai curries or scrambling up in place of eggs for recipes like breakfast burritos. Nuts last quite a while, so you can buy in bulk when they are on sale. Bread and peanut butter are also cheap (though I'd look for whole-grain bread and PB with little sugar, since you're concerned about health). Fresh or whole frozen fish and good cheese are expensive, and fishsticks and American cheese are cheap but not very healthy. Eggs are generally cheap, but full of cholesterol. It's a sad fact that the healthier things generally cost more. It's up to you how you want to balance all that out, but I'd suggest going light on the eggs and dairy, and making grains, veggies, fruit, and beans the bulk of your meals. My big mistake when I first went vegetarian was loading up with dairy and eggs to replace meat, and the consequences on my health were disastrous.

Rather than list a bunch of links for recipe sites, I'll give you a good hint. Google for "vegetarian recipes" and you'll get more recipes and whole recipe sites than you could ever dream to cook up in your entire life.

*About fish - if you choose to include fish in your diet, you have to consider mercury content. If you choose to exclude fish, you then have to consider getting enough omega 3 and 6 fatty acids, which you'd have to get from flaxseed, walnuts, etc. Either requires research - I exclude fish and think it's easier to get fatty acids from other sources than to worry about mercury, but it's your choice.