Q:

How exactly was Apollo CSM attitude changed (from a current one to a new required one)?

This answer to a related question suggests that the order in which rotations around three principal axes are applied (in order to estimate the conversion of an attitude to a set of roll, pitch and yaw figures) is important.
This leads to another question: How exactly were CSM maneuvers for the purpose of changing its attitude performed? 

Were there three separate consecutive sets of firings of the RCS thrusters (i.e. one firing to rotate around one axis, then firing to come to stop, then another firing around second axis, come to stop, then third firing around third axis, 
come to stop), or was there a multiple simultaneous firing of various RCS thrusters at once (i.e. to get to the the new required attitude in one go, so to speak)?
If it is the former, was there a rule for a specific order to be followed, for example, first Roll, then Pitch, after that Yaw, or maybe first Pitch, then Yaw  after that Roll?

Let's consider a particular example from Apollo 11 Flight Journal. Capcom Bruce McCandless at 025:49:20 into the flight gives PAD for Midcourse Correction burn number 2. The Journal editors give the following interpretation for the spacecraft attitude: 

Spacecraft attitude: Roll, 277°; Pitch, 355°; Yaw, 15°. This is with respect to the attitude of the guidance platform, itself aligned to the PTC REFSMMAT.

If I understand the quote correctly, then regardless of current CSM attitude, at the end of the maneuver, the spacecraft will have to be positioned at Roll 277°; Pitch, 355° and Yaw, 15° with respect to the corresponding axes of the current alignment of the guidance platform.
I assume the astronauts just plug in the numbers in DSKY, and computer does the calculation to get them from whatever their current attitude is to the required one.

A:

A single axis rotation (SAR) was performed from the initial to final attitude, unless they were avoiding gimbal lock, in which case it was split into two rotations.

It  has  been  shown  in  MIT/IL  Report  E-1832  that  it  is
convenient  to  perform attitude   maneuvers   by   simultaneous
maneuvers   in   three   axes. However,   under   certain circumstances
this  leads  to  maneuvers  through  the  area  of  gimbal  lock
warning  on  the Inertial   Measurement   Unit. In  this  event  the
maneuver  is  split  into  two  parts  such that  the  gimbal  lock
area  is  avoided.
The  inputs  to  the  attitude  maneuver  computation  are  the  three
gimbal  angles desired   as   the   final   orientation   of   the
spacecraft with   respect   to   the IMU  stable  member. In   order
to   convert   from   spacecraft   axes   to   control   axes,   all
outer   gimbal   angles   are modified   by   the   7. 25 degree
reaction  jet  offset.The  rotational  rate  to  be  used  by  the
maneuver   is   also   required.
From  the  present  gimbal  angles
and  the  required  final  gimbal  angles  a  rotation matrix   is   computed   which   describes   the   transformation   from   the   initial   to   the   final attitude. From   this   matrix   the  eigenvector   giving   the   direction   of   required   rotation is  derived   by   partitioning   the   matrix   into   its   symmetric  and   antisymmetric   components. The   equivalent   angle   of
maneuver is   obtained,and  using  the  magnitude  of  maneuver
rate,   the   time   of   maneuver   is   computed. In  addition,  the
rotation  vector  of  the maneuver   rate   is   resolved   into   the
three   control   axes   of   the   spacecraft.
The   inputs   from
the   maneuver   computation   program   to   the   Reaction   Control
System   being   described   are   these   three   spacecraft   rates,
and   the   time   of   maneuver. Where   gimbal   lock   is   to   be
avoided, the  two  component  maneuvers  are  sent  to  the control
system    separately.

Apollo GNC paragraph 3.6

