US Citizens Marrying Abroad?

Being married in Bulgaria is being married in the United States as well. You'd tell the US government about it by filing your federal income tax returns as married. Most likely you will file jointly with your wife anyway. Generally speaking, Uncle Sam doesn't care if you are single married, straight or gay.

Your wife would go to the SSA office and change her name based on the marriage certificate from Bulgaria, accompanied by a translation into English. Once she has received her new SS card, she can get a new driver's license, new passport (not really needed), bank ATM card, credit cards, etc. Since this is such a big inconvenience, many people don't change their name anymore when getting married.

A legally valid marriage in Bulgaria is legally valid in US or elsewhere. As for name change, if she wants her passport with her married name, she can go to US consulate with marriage certificate and apply for a new passport, or do it after her return (need official translation if marriage documents are in Bulgarian!). Use the marriage certificate & official translation to change name on Social Security Administration records (that's necessary!), driver's license, anything else she wants in married name instead of keeping her maiden name.

Best to change everything, or else keep maiden name on everything so all documentation has the same name. Avoid confusion!

a marriage in Bulgaria is valid in the US

it may be sensible o have a basic wedding in the US first

then another wedding in bulgaria fpr the relatives

Except for notifying Social Security of the name change perhaps changing her drivers license, nothing else should be required.