Most music suitable for dancing has an introduction of a few bars (often 4 or 8), and you should use that time to listen for the accented beats and the musical phrasing. Counting won't really help - you have to hear the music, and it will tell you the moment to start dancing.
In most of the dances, the man can choose which foot he leads of on: left or right. He needs to let his partner feel his weight distribution, and ideally, if he uses the introduction to shift weight from one foot to the other (with very little actual body movement, the lady should feel the shift) and will respond by changing her weight too. A useful tip would be to make those weight changes correspond with the accented beats in the music, so that both of you are already moving in time with the music (although probably not enough for anyone else to see).
When you move off, swing the leg so that the foot is placed on the floor on the beat, and allowing for musical interpretation that might then depart from strict rhythm (particualrly as you become more advanced), each step will fall on a beat, or if you are dancing a syncopated pattern, on an identifiable fraction of a beat.
Stop counting as soon as you can: listen to the music instead, and dance in response to it.