Basic Boeing philosophy on the old types (737) - use AT with AP engaged. With AP disengaged, disconnect AT.
The automatics on the 737 are not top drawer though - the AT needs "help" in gusty conditions. The thrust levers move according to the AT demands.
The modes available for the 737 NG AT:
N1 - AT holds commanded N1 limit. Used in takeoff and climb, with aircraft pitching for speed
MCP SPD - AT holds target speed set on the MCP using thrust up to current N1 limit. Used in level flight, approach mode and in climb/descent when V/S is the active pitch mode
FMC SPD - AT holds target speed from the FMC - normally in cruise or in a path descent on a geometric descent path with VNAV PATH as the pitch mode
ARM - thrust levers static and able to be positioned manually. Normally idle thrust in descent with MCP SPD as the pitch mode (Level Change) Also provides very limited low speed protection.
GA - provides reduced thrust if the TOGA button is pressed once on a go around equivalent to around 2000fpm climb rate
It's a fairly simple system, with the main point being that if you are flying manually, you should be using manual thrust. I have never tried manual flight on approach with the AT in, and I imagine with the pitch couple it is fairly interesting.