There are basically two different ways to feed corals, broadcast feeding or target feeding. Broadcast feeding is simply adding specialized food into a high flow area of your tank usually no more than once a week.
Target feeding is done with a syringe and (often with a plastic tube connected at the end of the syringe to get into deeper, less accessable areas of a tank). When Target feeding - it's important to turn off any powerheads/wavemakers so the water's as still as possible. Personally I also turn my protein skimmer off and leave my return pump running. Only need to do this for about 10 min after feeding the last coral.
After you've put the proper amount of food into the syringe, release food approximately 2"- 4" upstream of the coral so the food naturally flows into the polyps.
It is very easy to overfeed when target feeding. Overfeeding, as you know can create a whole other mess when it comes to water chemistry, algae blooms, etc so please stick to only recommended amounts of whatever food you are using.