Where can my Albanian husband travel abroad on honeymoon without any problems?

at the Albanian Ministry of Foreign Affairs webpage there is a list of the countries in which you can enter without a visa. It's a short list though. For the others you have to apply for visa at the local embassy or consular section.

http://www.mfa.gov.al/

Albania.

What sort of problems are you expecting? Otherwise we can't tell you how to avoid them.