Assuming you've made a stabilized approach (constant airspeed, stable rate of descent, and without the need for major control inputs) then the flare is just a matter of practice. I imagine as you get close to the runway (say roughly the same height above ground as a hangar roof) you will gradually reduce power to idle and pitch up from your nose-low approach attitude to a more nose-level attitude. This is often called the round-out and immediately precedes the flare. When I talk about a flare I am assuming you are already just a few feet above ground.
The flare, as you correctly stated, is the time when you gradually pitch up as the aircraft slows without climbing (ballooning). The purpose is to slow the airplane down and touchdown at the slowest possible speed. The difficulty you have with making the flare without ballooning is very common, and no matter what it will take practice. The best way to practice this is to sharpen your sense of vertical movement while landing. I recommend looking further down the runway--toward the horizon, and be aware of cues in your peripheral vision that your airplane is sinking or rising. Your peripheral vision is particularly useful when the aircraft nose blocks the horizon. Pay attention, and over time you will develop a feel for the aircraft's sink-rate.
In the flare you should increase pitch to keep the airplane barely sinking toward the ground--we're talking almost level. As the airplane slows in airspeed you will feel it sink. Increase pitch to all-but-stop the sinking. If you increase the pitch to the point where the airplane completely stops descending, then hold that pitch where it is until the airplane starts to sink again. If you pitch up too rapidly and the airplane starts to balloon, your best bet is usually a go-around, although very very slight balloons can sometimes be corrected by simply holding the pitch and waiting for more speed to bleed off and the airplane to sink again. In all cases, resist the urge to lower the pitch of the nose. Your airplane is still slowing down on its own, and lowering the nose will kill what little lift you have left and could set you on the pavement pretty hard.
The rate at which you'll have to pitch up in the flare will vary depending on the day, the weight of the airplane, your airspeed, winds, etc... Some days it will be very fluid and gradual, and on other days you may find yourself having to make a few quick pitch adjustments. But if you try to make your adjustments in response to how rapidly the aircraft is sinking toward the ground based on visual cues out the window, then you should have good luck.
Remember you a trying to control a one ton piece of aluminum flying at 50 mph within the precision of a few inches... it definitely takes practice! Have fun and good luck!
coaching over 3000 student landings