Hi Darryl:
Assuming your tub is a fiberglass unit, the Hardibacker under the tub would be fine, but not necessary. However, you should put something under the tub when you install it to stiffen up its floor. I've seen plaster or drywall compound used, but these tend to shrink. They make a special mortar product just for this purpose that sets fast and doesn't shrink. I forget what it's called (underlayment cement?), but you just mix a 25lb bag of the powder with water and trowel out a good sized blob under the center of the tub unit. When you set the tub onto the mortar, it conforms to the bottom of the tub and really stiffens it up once it sets.
Given a choice, I use screws over nails every time. They tend to hold better, are easier for me to set to the correct depth, and are more easily removed if necessary (though I realize this may not be a consideration in your situation).
Don't worry about this being your first tile floor. It's not that hard to do. Just take your time laying them out and cutting. As long as you have a sturdy subfloor and underlayment, it should last a long time.
Regards, John.