In answering your question, my first thought was to look on-line for some good goat meat recipes. So, do an internet search yourself and see what you can find. After a year in Afghanistan with the military, I realized that there are a lot of good ways to cook goat and if done properly can be rather tasty.
Then, of course, I figure it is important to let you know that you cannot harm these animals just because they come onto your property. More than likely, you could end up in jail. So here is what you do.
First of all, give your neighbors notice in writing via certified mail telling them that their animals are coming onto your property and causing damage. Give them details of your previous efforts to solve the problem and relate the damages to date. Inform them that they will be responsible for any and all property damage. You will need to keep a copy of the letter and the return mail receipt.
Then, every time the goats come onto your property, call your local sheriff's department and animal control. Most states have laws about animals not being under the owner's control. Additionally, most counties have animal control ordinances which include fines and the eventual destruction of the animals. Eventually, your neighbors will get tired of getting animal control citations and having their animals taken to the local animal shelter.
You also need to document all property damage caused by both the animals and the children next door. The best way to do this is to take before and after pictures of the fences and plants.
If there has been property damage and you have evidence of such, you then need to get a written valuation of the damages. Normally, this is done by an expert who repairs or replaces the damaged items. Then you can file a lawsuit in small claims court.
However, above all else, you need to consult with and hire a local attorney to discuss your rights and options further.
I agree with my colleague. I only add that you should take pictures/video of the goats and children on your property while you are calling the police/animal control, preferably with a camera that has a date and time stamp.
If we do not have a signed fee agreement I am not your attorney and this is not legal advice.