When you start a turn to the right, your right ski is the inside ski. We use the terms inside and outside ski because these terms are the least confusing. We could talk about downhill vs uphill skis or left vs right, etc. Each approach has it's own problems.
There are many different ways to start turns. Whether you tip first or turn first or weight first or make some combination of moves first is a matter of choice for a skilled skier. Inexperienced skiers have fewer options. Skilled skiers balance over their outside ski. For a right turn, this would be balancing against the left ski. Whether you purposely shift your weight onto the left foot or wait for the forces involved in turning to the right to "pull" your weight onto the left foot is a choice you can make. Opinions on which approach is better vary. Try both and decide for yourself.
Skis are shaped fat at the ends and skinny in the middle. When you tip a ski onto an edge, it creates an arc in the snow. If the ski is pressured so that the edge can grab in the snow, the ski will want to travel along the shape of the arc (i.e. turn). If the ski is not pressured at the tip or the tail it will not want to travel along the arc very well. Because a ski is typically moving forward in the snow, most skiers don't have any trouble pressuring the tail of the ski. Many skiers do have trouble moving their body to stay with the ski (i.e. the skis tend to "pull" them down the hill). This makes it hard to maintain pressure on the tip of the ski. A mental focus on the toes often solves this problem.