What US customs procedures must the pilot of a private aircraft follow when traveling in and out of the country?
General  


Each person leaving the country must be authorized to return to the US.  (ie. they are a US citizen or their visa allows them to leave and return to the US.)
Ensure that you have all required paperwork

Pilot license
Medical
Aircraft Registration
The aircraft must have a current customs decal.

Ensure that the aircraft is properly equipped

12" Registration Numbers
Transponder



Before Leaving the US  


File eAPIS

eAPIS is required to be filed at least 60 minutes prior to your departure.
Other third party services (ie. FlightPlan.com, etc.) may make this easier if you do it a lot.

File an international flight plan

May be an instrument or a defence VFR flight plan



Before Returning to the US  


File eAPIS (see above)
File a flight plan (see above)
Notify customs and obtain permission at least one hour prior to landing in the US.  Record the initials of the customs officer that you spoke to just in case they misplace your notification.

Note that if you are coming from locations "south of the US" (for the specific definition, see here), you must stop at the designated airport closest to your point of border crossing to clear customs (unless you have an overflight permit).



Upon Arrival in the US


You must land at an airport of entry.
Report immediately to the designated customs location
Do not allow anybody or anything to leave/enter the aircraft until the customs official gives your permission

The only exception that I know of to this are the South Florida airports where you are expected to take all people and belongings into the customs facility for inspection.

Discard any international garbage in the approved manner.
Provide required paperwork

A Customs Declaration for each arriving family

Wait for customs to specifically tell you that you are cleared before you attempt to leave.


Notes

This is not all-encompassing, please help add to and expand this list so that others may learn from our experience!

A TON of useful information is made available by the FAA in the Guide for Private Flyers.