To let down the mainspring the movement will be required to stay in the case (pocket watch movements) or will require the stem to remain attached (wrist watch movements generally don't have to remain in the case to use the stem). When the stem is advanced slightly, the click (the ratchet that prevents the mainspring from unwinding) will creep along the largest gear on the top of the movement, which is connected to the mainspring barrel. It is when the click is advancing that a small pin or tool can be used to disengage it from the mainspring - be very careful to hold onto the stem tightly and release energy carefully. If you lose your grip on the stem and have the click held open, then the mainspring will make a 'snapping' sound as it unwinds in an instant - this can damage any number of parts in the watch, such as the going train (gear teeth, pivots, jewels) and very likely the mainspring itself.
Once the mainspring is let down the dial will need to be removed. Typically this is achieved through the releasing the dial foot screws located on the side of the movement. In older pocket watches such as this Elgin, you may find three screws to release. On most wrist watch movements, there will typically be two screws holding the dial to the movement. On this Elgin, the dial screws are at the 4, 7 and 11 o'clock positions.
The dial can then be removed to expose the obverse of the movement.
The balance can then be examined, check to see if the jewel, spring, pivots, poise, trueness, etc. are all good.
Before the mainpring barrel bridge can be removed, the winding gear on the top of the mainspring must be removed, along with the transmission gear:
The two screws holding the bridge to the plate can be removed, and the bridge can then be lifted:
With all the main bridges removed the movement should look like this:
Time to get that mainspring barrel out of there:
And then the careful removal of the wheels, don't hold the wheel by the teeth/pivots:
Now the cock holding the pallet fork can be removed, along with the pallet fork:
I've left the keyless works to the end of the disassembly. There are actually four small individual components which make up the clutch pillar - being removed in the following two photos:
I actually don't know what this part being removed is called, it assists with the selection of the winding/setting for the clutch pillar. The arm at the base of the clutch pillar also needs to be removed (no photo for this), it is simply held in place by a screw.
Now the movement is flipped over and the hour wheel can be removed:
The bridge holding the minute wheel can be removed. The two screws holding the bridge are quite small, so care should be taken when holding with tweezers - if you apply too much pressure and the screw pops out of the tweezers, you'll have almost no chance of ever finding the screw again....
The cannon pinion can be removed, another tool is required to lift it off the centre wheel, but I didn't get a photo of that.. so here's me holding it above the centre wheel..
The bare main plate stripped down apart from the balance jewels, and the pins for the pallet lever:
The (mostly) disassembled movement ready for cleaning: