Which altimeter setting should I use?
Yes, that is correct procedure, except that if you're flying an instrument approach, switching to destination altimeter should be part of your approach setup "chart briefing" (which likely happens before you get handed off to the tower).

From AIM 7-2-2(a)(1) (emphasis mine):


  [Set the altimeter] to the current reported altimeter setting of a station along the route and within 100 NM of the aircraft, or if there is no station within this area, the current reported altimeter setting of an appropriate available station. When an aircraft is en route on an instrument flight plan, air traffic controllers will furnish this information to the pilot at least once while the aircraft is in the controllers area of jurisdiction. In the case of an aircraft not equipped with a radio, set to the elevation of the departure airport or use an appropriate altimeter setting available prior to departure.


Unfortunately the AIM text doesn't directly address destination altimeter settings, but section 4-1-8 alludes to the fact that controllers must verify you have the best destination altimeter setting, either by telling you or by having you report that you have the current ATIS or the AWOS/ASOS broadcast ("the numbers" for the latter):


  a. Numerous approach control facilities have established programs for arriving VFR aircraft to contact approach control for landing information. This information includes: wind, runway, and altimeter setting at the airport of intended landing. This information may be omitted if contained in the Automatic Terminal Information Service (ATIS) broadcast and the pilot states the appropriate ATIS code.
  
  b. Such information will be furnished upon initial contact with concerned approach control facility. The pilot will be requested to change to the tower frequency at a predetermined time or point, to receive further landing information.


For instrument approaches under IFR, setting and checking the altimeter to the destination's setting should be part of your approach briefing checklist. My personal "checklist" includes the altimeter setting as part of the weather information when I get to the bottom of the chart and check ATIS/ASOS/AWOS numbers against the approach minimums.

Interestly enough, the FAA's Instrument Flying Handbook doesn't have much to say on altimeter settings at all (except for mentioning that keeping the setting up to date is important and that bad things can happen if you don't).