There's a tendency for some skiers to use the kick start that they see the pro racers use, but end up kicking their feet ahead of their upper body. When this happens, there's a stall transitioning from the initial push to the skate. If this is the problem, my recommendation would be to focus on using your poles during the initial push to keep your upper body ahead (and leaning forward) of your lower body. You have to hold your poles in the snow through the kick so that they are angled forward as you complete your push off them.
Skating is a drill we do to improve skiing skills. Hmm - this is a toughie - how do we do a drill for a drill?
The ley to skating is rolling onto the new edge before push off so that the upper body is already moving in the direction of the other ski before you shove off onto that ski. One weird drill for this is tractor tire. Find an almost flat trail that has snow soft enough to leave tracks in. Get in a wedge and go very slow but have one ski flat and one ski on edge. This causes your wedge to "crab" sideways a bit. Travel for a ski length, then abruptly switch the edged to ski to flat and the flat ski to edged. This will cause you to crab back in the other direction. The flat skis smears the snow and the edge ski leaves a straight line. After a series of these, you will have left straight lines in the snow that connect perpendicular to each other that will look like tractor tire tracks in the snow. This drill helps to isolate the same movements in skating but at a much lower intensity.
A drill that might help with "transitions" is skate to shape. Find an easy blue trail and start skating straight down the fall line and gradually turn the skate moves into short radius turns. The smoother you can make the transition so that it is hard to tell where the skating stops and the turning begins, the better your starts are going to be.
Does this help?