Slightly different with a full floating line vs. a sink tip of some sort.
Dry line: Cast across and slightly down stream and one or two up stream mends (best if the first done as an 'air mend'
). Object is to get the line above the leader and fly. Whole idea is to get the line up stream from the leader. Current will normally pull the fly ahead of the leader.
Doing an 'air mend' takes a bit of practice, but not much. If the the line lands on the water, still the one/two mends BUT with your rod tip as high as you can get it as you want to 'lift' the line as you move it up stream. A 'drag' is a no-no. Think of an upside soup bowl.
Same idea as above but aim your cast further down stream (I usually go for about 15%, your mileage may vary. A single up upstream mend is usually enough to bring the line under tension. Job done.
But like all 'Rules' sometimes you have to break them. Example you have a 'wide' section of far slower water you have to cast over to get to the 'Holding water.' There your mends will be DOWN STREAM. Follow the fly with your rod tip and flick the line in the slower water to keep things lined up. If you don't your line will end up in a "L" shape and no way can you set the hook.
Hope all that makes a bit of sense.
Minor edit: Do you hold back with your rod tip or rod tip slightly ahead of your line? Usually the first will slow up your swing, leading will speed it up.