Examine the levers and buttons on your integrated shift/brake levers. Get accustomed to the basic principle behind their use as well as the multitude of hand positions available on the bars. Locate the brake levers, which are the largest levers on an integrated brake/shift system. The brake levers are activated by pulling them backward toward the curve of the handlebars.
Locate the down-shift lever on the right shifter. On Shimano STI shifters, the down-shift lever is activated by pressing the brake lever sideways, toward the front wheel of the bike. On Sram and Campagnolo shifters, the down-shift lever is located behind the brake lever, activated by pressing it fully toward the front wheel.
Locate the up-shift lever on the right shifter. On Shimano STI shifters, this is the small lever behind the brake lever, activated by pushing it inward toward the front wheel. On Sram shifters, the up-shift lever is the same as the down-shift lever. Pushing the lever fully (two clicks) shifts down a gear. Pushing the lever only slightly (a single click) shifts up a gear. On Campagnolo shifters, the up-shift lever is located on the left side of the shifter body. It is activated by pressing it down with your thumb.
Examine the left brake/shift lever. The left lever works exactly opposite from the right shifter. Pushing inward on the down-shift lever fully causes the bike to shift up one gear. Similarly, pushing inward (on Shimano and Sram shifters) or down (on Campagnolo shifters) on the up-shift lever causes the bike to shift down one gear.
Take your bike out for a ride. Start in a small gear, one in which it is easy to pedal. When you are ready to shift up to a higher gear, press the up-shift lever of the right shifter. Continue until you can not shift up any further. When this happens, press the up-shift lever of the left shifter once, and the down-shift lever of the right shifter a few times. This will allow you a few extra gears to accelerate in. Continue in this fashion until you are ready to stop, at which point you should pull backward on the brake levers to activate the brakes. With practice you will become smoother and more efficient in shifting and braking with integrated brake/shift levers.