As a US citizen living abroad, do I need to register anywhere that I got married and had children?
Do my children automatically get US Citizenship?
You don't need to register your marriage, and you don't have to register your children. Children of US citizens born abroad are considered citizens, however if you want it to be official and for them to get passports then you will need to go to the embassy with your children (having your spouse as well is a good thing), along with the forms and proofs needed for a consular report of a birth abroad, passport application and photos (make sure you get them in the right dimensions!), and social security card application.
Every embassy has a web page that will show you the process, you can find yours here.
Some countries require that you obtain a clearance from your embassy if you wish to marry a citizen of that country. The Philippines is one such country. When I received my clearance, I took an oath in front of a consular that I was not married, and possessed the full legal capacity to enter into a marriage agreement.
They'll issue you a notarized affidavit, which you present to the agency that controls marriage licenses in the country that you reside. Not all countries require this.
Regarding registration, you aren't required to notify them of marriage, but you (yourself) should register with the embassy in the country that you reside and keep your contact information with them on file up to date. If something happened that they needed to alert you (political unrest, a threat against US citizens, etc) - they should be able to get in touch with you.
You should also report the birth of your child to the US consular as soon as it's convenient for you, your spouse and your child to make your way to the embassy. Most embassies will ask that you call ahead, make an appointment, and will give you a list of requirements. All of you must be present, with everything on the list of requirements, or the process could be delayed quite a bit.
The paperwork will be processed, and your child will get his or her passport and consular issued certificate of birth (report of birth abroad) usually via courier.
Keep the following in mind
In order for your child to be able to transmit their citizenship to their children, they must reside in the US for four years contiguously after their 14th birthday. Additionally, for you to be able to transmit your citizenship to them, you must have done the same.
So yes, to answer your question, it is automatic, provided that you meet the requirements set out by Congress in order to transmit your citizenship. However, your child will not be able to enter the US without a passport, so you'll need to get them set up.
As a note for statutory requirements for citizenship, you will want to register any children born abroad, as the other two posters mention, as soon as reasonably possible.
The actual requirements are actually a bit complex. In general if both parents are US citizens, then the child is automatically a citizen if either parent has ever resided in the US.
If one parent is a citizen and the other is a non-citizen (ignoring for the time being non-citizen nationals which would be a case unlikely to be relevant unless travelling abroad with a national resident of US Marshall Islands or the like) then the rule is 5 years residency/physical presence, at least two of which have to be over the age of 14. I didn't see anything in the statute relating to citizenship at age of residence, and I doubt that courts have had a chance to adjudicate the issue.
Consequently I don't think anyone can say anything beyond the argument that the simplest way to read the statute is that it is 5 years of physical presence in the US as a resident. The government and courts might disagree if such a case actually came before them however.