You may travel abroad after you file your U.S. citizenship application. Your Green Card will allow you to make short trips. However, you need to plan your trip in such a way that it does not affect your biometrics appointment and your naturalization interview. U.S. Citizenship and Immigration Services (USCIS) generally takes around 6 months to process a U.S. citizenship application and you are likely to receive your biometrics appointment notice within one month from the date of your application. Hence, it is wise to appear for biometrics and then travel abroad. Likewise, you need to return to the country to appear for the naturalization interview. When you apply for citizenship through naturalization, the English and civics tests are important.
Generally, a foreign trip after filing Form N-400, is not recommended. Nevertheless, if you have to leave the country to undergo medical treatment or for some other reason, you may leave the country. In such cases, you need to check your mail often or you can ask your family member in the United States to check your mail and inform you about your interview notice. You must not miss your biometrics appointment and your naturalization interview and you will get into trouble if you miss your biometrics appointment or if you fail to appear for the interview.
If you cannot attend the interview on the scheduled date, you can request the USCIS to reschedule the N-400 interview date. However, your interview will be rescheduled only if the reason is genuine. You might receive your naturalization interview letter within 3 or 4 months from the date of your biometrics appointment. During that period you may travel abroad but see to it that you get back to the country as soon as possible. That is because your naturalization application will be denied if you miss the interview. Hence, it is wise to check your mail regularly. If you receive an appointment notice from the USCIS, you may cut short your travel and return to the U.S and report for the U.S. citizenship interview. Moreover, you need to remember that if you remain abroad for 6 months and more, you will become ineligible for naturalization.