A step increase in elevator (which is then held) will first upset the pitching moment equilibrium of the aircraft -- the pitching moment will be non-zero. It will pitch nose up, increasing angle of attack. However, this increase in alpha will over-shoot the new equilibrium trim point and the short-period oscillation will occur. This will damp out quickly (on the order of a few seconds).
At that point, the aircraft will be flying at a new alpha and CL, but will not yet be at the equilibrium velocity for that CL. The aircraft will be in a slight climb, speed will quickly bleed off while it gains potential energy. This will over-shoot the new equilibrium velocity and the phugoid oscillation will occur. The aircraft will trade kinetic for potential energy until drag damps out the excess energy and a new equilibrium velocity corresponding to the lift coefficient is achieved. The phugoid will damp out over the next 30sec to several minutes. Aircraft with excellent L/D will take longer for the phugoid to damp.
Since nose-up elevator was input, this new CL will be higher than CL0, and the equilibrium airspeed will be lower. The drag coefficient will be lower, but whether the drag (or power) is less depends on whether the initial equilibrium point was above or below the point of minimum drag/power i.e. were you on the front or back side of the power curve?
Here, we will approximate propeller powered aircraft as having approximately constant propulsive power with velocity -- and we will work in terms of the power curve. We would approximate jet powered aircraft as having approximately constant thrust with velocity -- and we would work in terms of the drag curve.
If you are on the front side of the curve, your increase in CL will drop Velocity and will increase CD but will lead to a reduction in drag/power required. At constant throttle, the aircraft will start to climb (as thrust is now greater than drag).
However, if you are on the back side of the curve, your increase in CL will drop velocity and will increase CD -- but drag/power required will increase. At constant throttle, the aircraft will start to descend (as thrust is now less than drag).
If, instead of holding the new elevator input constant, you were to simply pulse the elevator and let it return to its neutral position (assuming you were flying stick-free trimmed at the start).
The initial reaction would be similar -- the short-period would be excited, but would quickly damp out -- this would excite the phugoid, which would take longer to damp. At the end of the phugoid, you would return to essentially your initial velocity in steady level flight. Your altitude might be slightly lower or higher than your initial altitude. You would not be in a significant climb or descent -- you should return to your initial equilibrium condition.