First of all, color everything before assembling. If you're good at shading, try your hand at shading a bit centers of the petals. Let dry.
From what I can tell, you glue the smallest piece to the one in the middle of the three on the left. Try to arrange it so that comes out looking like the petal that has the roundest shape. You'd have to use the punch twice so you get 2 of each of the other two petals. So you'd need two of the first and third on the left. The first of the 3 of the left go on the upper part of the orchid. Think of it as if it were a star shape. These two petals would go one on each side. The lower petals of the orchid would be made by the other two petals, the ones that are the third shape on the left of the punch.
For the center, color front and back before assembling. After you have colored, roll up the lower part of the fleur de lis-looking punch shape. The part with the three areas on one side points up. Before permanently attaching this part, you might first want to roll it up and play with it until it looks like how you want it to look, and make sure the coloring is correct on both sides of the piece of paper.
I suppose you should use very strong glue or tape. If using glue, test a piece of paper you have already colored to make sure the coloring doesn't bleed or warp.
It looks like the person glued the petals on to a piece of paper that was the same color as the petals. If you look closely, you'll see a gap between petals that shows yellow, when in real life that space would be an actual gap. That might be easier than assembling on thin air.
This is how the orchid is supposed to look like. Notice how the middle spotted part points downwards. The roundest petal, which would be the top spike of a star, points up.