After you finish the game once, press any button on the Title screen and the cursor will change from a mushroom to a Mario face. Press Start to begin the expert level. You can play through again having more enemies, harder enemies, and enemies in different places.
Extra Coins and Part of Level Skip
On the third level, at the very beginning, there is a wall and a small "walk under" area to which you can not go. Before the "area" jump at the edge of the wall, a little thingy should appear. Go onto it. It will go up past the ceiling and you will be above a small section of the level and also find some coins.