If flaps are used on takeoff, when should they be retracted?
The short answer is, it depends. The FAA Airplane Flying Handbook merely states that this can be performed once the aircraft has established positive climb, but does not specify exactly when.

Ultimately two situations arise:


The airplane is sufficiently large/high performance that normal operating procedures specify the use of flaps for all takeoff scenarios, in which case the manufacturer or the operator (if applicable) may well specify a flap retraction schedule, which seeks to allow the aircraft to accelerate effectively during climb out, avoid overstressing the flaps by having the aircraft fly past the maximum speed for each flap setting, while at the same time maintain a comfortable margin above stall speed.
The airplane only requires flaps for certain kinds of takeoffs such as maximum performance or soft-field efforts. In that case, procedures may be somewhat looser in defining the exact moment of flap retraction. A climb checklist, should you choose to use one, will likely mention the flap configuration as an action/verification item, so that would likely be a good moment to retract them if you haven't done so already. Personally , I usually retract them passing through 200-300 feet AGL, which usually coincides to the aircraft being at least half-way between clean configuration Vx and Vy (I've been mostly flying Skyhawks, Skylanes, Arrows, and Huskies). Whichever way you do it though, you want to avoid retracting them too low and too slow, but also forgetting about them and blowing through Vfe (which even light, slow airplanes can do quite handily).


With respect to the actual changes in aircraft behavior, you'll likely notice the following things:


A slight-nose down moment as wing's characteristics change, which influences the airflow reaching your horizontal stabilizer. The converse applies when extending flaps. Mind you, this is an airplane model-specific thing and one that seemed most noticeable to me in the Cessna family of light singles. 
For the same power setting and pitch attitude, the aircraft will begin to accelerate quicker for each notch of flap retraction.
Climb rate is trickier as it is dependent on your power setting, airspeed, and pitch angle; your inputs will matter. However, generally speaking, the reduction in drag would lead to a greater difference between power available (or power set) and power required, so, all things being equal, you would likely see an increase in climb rate.