I stayed a long time in rural Aceh, so here are my tips:
First of all, do not think of Indonesians as very religious. They are usually traditionalist (some may say conformist). This is different.
You should both wear pants below the knees and real shirt (no tank top), no cleavage, no belly button displayed. Nobody will throw rocks at you if you don't, but they will be more friendly if you do.
Do not display signs of affection
Do not kiss in public. Even holding hands should be avoided (same-sex couple, see below). Once again, nobody will throw rocks at you but...
Unmarried in the same hotel room
This is legal only if BOTH of you are foreigner. This is not only Aceh, but the whole Indonesia.
If one of you "looks like an Indonesian" expect problems when booking a room. My advice: pretend you are married. If it gets complicated give them any text in the Cyrillic alphabet and say it is a marriage certificate (same-sex couple, see below)
This is also true in the mostly Christian province of North Sumatra (booking a room should be easier.)
In a "tourist Ghetto" nobody will have a problem if you drink booze naked in the street (after all, you conform to their idea of a western environment.) But as soon as you go to the nearby village, conform fully to Indonesian standards. Even a bartender or a gogo dancer would be tolerant towards a foreigner, but very strict at home.
Same sex couple
Counter intuitively, same-sex couples may have it easy in Indonesia.
Friends (male or female) often hold each other in a way that would look romantic in the western world. Also you can book a room for 2 men/women with no question asked.