As you can see, the left hand is slanted slightly downward and the right hand is either parallel with the chanter or the fingers slant slightly downward. If you're fortunate enough to have a teacher, they can set you straight, otherwise, study the images in your piping tutor book or those on the Web.
When practicing, rest the chanter's bottom on a table or your leg, with your sheet music easily visible. I sometimes put it on another chair, and end up crouching over the music, with the chanter vertical (which echoes the position of the bagpipe chanter. It's also possible to play the practice chanter standing. The key here is to be as comfortable as possible, both to facilitate breathing from the diaphragm and a relaxed finger position.
Try to avoid clutching the chanter in a death-grip, even though you have been previously admonished to have your fingers be as straight as popsicles. Clutching the chanter like this not only tires your fingers and wrists unnecessarily (possibly leading up to carpal tunnel syndrome) but is a BAD habit to get into. If your fingers are unnecessarily tense, it will be difficult to do grace notes with any speed or grace (pun unintended). To quote an unnamed piper, you should hold the chanter "as though you were holding a piece of [soft] human excreta". Quite an image. A good exercise to do before practice and whenever you become conscious that you have a choke-hold is to hold your hands in front of you and shake them, even flop them about, from the wrists. The resulting finger posture is what you should adopt on the chanter.
A short note on practice. Practice early; practice often. It is better to have four fifteen minute sessions than one hour session. Mind that you aren't practicing bad habits. To quote the Peking Piper, "Practice doesn't make perfect; perfect practice makes perfect." Finding a private place to practice will help you avoid that fear of appearing foolish. (You may eventually want to get over that sort of stage fright by practicing near a supportive ear. You'll eventually hear comments like "you've improved", etc.)
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