For a steady state flight, there is a equation that FPA + AOA = Pitch, but how these three variables are correlated during a flight maneuver like currently I am in a leveling flight and then I want to change the flight path angle to 5 degrees, can I assume that AOA will be unchanged during the maneuver, so I could simply pitch up 5 degrees more to get a 5 degree FPA as delta FPA will be same as delta pitch（ assuming the speed is well maintained thru the thrust）?
The required lift force is about the same in a shallow climb as in level flight. (Strictly speaking, a steady-state climb always requires less lift than level flight-- for more see Does lift equal weight in a climb? -- but the difference is very small for climb angles up to about 30 degrees. At a 5-degree climb angle, lift is .996 * the aircraft weight.)
The amount of lift a wing generates at any given moment depends on the wing's angle-of-attack, and on airspeed. Lift is roughly linearly proportional to angle-of-attack through much of the flight envelope, and also is proportional to airspeed squared.
Therefore the airspeed indicator can be thought of as an angle-of-attack indicator, at least during non-accelerated maneuvers, i.e. during 1-G flight, excluding very steep climbs and dives.1 If the airspeed has decreased and the aircraft has remained in 1-G flight, the angle-of-attack must have increased.
Also, to a first approximation the control stick or yoke is really an angle-of-attack controller, and the fore-and-aft position of the control stick or yoke serves as an angle-of-attack indicator. For much more on this, see Wolfgang Langewiesche's famous book "Stick and Rudder". Generally speaking, if the pilot moves the control stick or yoke aft, the angle-of-attack of the wings is increased. Of course, thrust changes can produce changes in pitch trim that complicate this relationship-- in some aircraft the angle-of-attack will tend to increase, or decrease, when thrust is increased. (Also, increases in G-loading make an aircraft tend to trim to a somewhat lower angle-of-attack for any given position of the stick or yoke, due to aerodynamic effects generated by the curving flight path. This effect is most pronounced in relatively slow-flying aircraft, like sailplanes.)2,3
So generally speaking, unless you are making a specific effort to ensure that the airspeed stays constant (e.g. by adding power as needed), you can't assume that when you pitch the aircraft up 5 degrees, the angle-of-attack of the wings will stay constant, and that's why you can't assume that you'll get a 5-degree change in the angle of the flight path. On the other hand, if you are keeping the airspeed constant-- which obviously would require an increase in thrust and power as the climb is entered-- then this would be a fair assumption.
Or more broadly, we can say that the airspeed indicator can be thought of as an angle-of-attack indicator in flight at any given constant non-zero "measured G-loading", where by "measured G-loading" we actually mean only the G-loading component that acts in the "upward" direction in the aircraft's own reference frame. Essentially, what we read on a G-meter mounted on the instrument panel. Note that this "measured G-loading" would be zero in a steady-state vertical climb or a steady-state vertical dive, as well as in a parabolic "push-over" maneuver.
This tendency to trim to a lower angle-of-attack when the flight path is curving in the "upward" direction in the aircraft's reference frame, can also be described in terms of "pitch damping", i.e. an aerodynamic "resistance" to pitch rotation.
Another factor that complicates the relationship between the fore-and-aft position of the control stick or yoke, and the resulting angle-of-attack of the wing, is pitch rotational inertia. To initiate a pitch rotation, the tail must generate a net pitch torque, above and beyond whatever pitch torque is needed to balance the wing's pitching moment. But we don't need to worry about this if we're just comparing steady-state cases.
FPA + AoA = Pitch
What is FPA? FPA is flight path angle. This is (in modern airliners) a way of autopiloting a descent based on angle rather than rate for a given heading.
So, pitch to the horizon is FPA + AoA, or FPA (where you're going) is Pitch - AoA. This part is correct.
assuming the speed is well maintained through the thrust
Thrust is the key. If your plane is staticly stable, adding or subtracting thrust will change pitch. Any changes in AoA will be transient.
If you are more staticly nuetral (depending on your CG location), pitch change is "helped" with a bit of yoke too (or you'll simply speed up when more thrust is applied).
After completing the manuver, for the same speed, AoA (trim) will be the same$^1$, FPA and Pitch will be different.
can I assume AoA will remain unchanged during the manuever?
Actually, no. During the manuver pitching motion and velocity changes will affect the angle of attack and lift output, which lead to deviation in the flight path. The best example of this is a failed loop. As the pitch goes to towards vertical, airspeed cannot be "well maintained" with thrust. The nose drops below the intended line of flight until enough speed is recovered to resume linear flight. If the plane stalls, the deviation is even more dramatic.
Conversely, a gentle increase in pitch (with a careful eye on airspeed) will lead to much gentler (if any) pitching oscillations. The plane will simply nose up and "climb" above its previous flight path. If done in a leisurely airliner manner (don't spill the drinks), AoA will not change much.
pitch controls speed, power controls altitude.
Or maybe, trim controls speed, power controls pitch. But if you are staticly nuetral, these functions are not strictly separated, and the aircraft will not seek out a given speed unless (you guessed it) it is programmed to.
$^1$ for a 5 degree climb, Lift requirement from wing (AoA) will be a hair less because the thrust is also doing some of the lifting against gravity.
AOA is useful information to have, but a great many aircraft do not have AOA indicators. It becomes more important as you explore the edges of an airplane's performance envelope, but for the purpose you describe it isn't necessary to know what your AOA is.
...in a leveling flight and then I want to change the flight path angle to 5 degrees, can I assume that AOA will be unchanged...
No, if you change pitch the AOA will change.
...so I could simply pitch up 5 degrees more to get a 5 degree FPA as delta FPA will be same as delta pitch（ assuming the speed is well maintained thru the thrust）?
Yes, depending of course on airspeed and pitch rate.
For the example you describe, if you are at cruise airspeed and want to begin a gentle pitch rate to a 5 degree climb, the change in AOA is negligible and virtually undetectable. You should be confident in programming your autopilot to perform gentle and gradual cruise maneuvers such as this without worrying that you lack AOA input.
In fact, I was trained to climb in either pitch hold or IAS hold, and descend in VSI hold. While I’m sure there may be airliner autopilots with AOA hold capability, I don’t see that feature, (or even AOA inputs) being necessary on a home built or experimental GA aircraft.