I had the same problem with my ipad. I bought a screen protector from Zagg hopping that it would make it a bit better. But it was even worse with the orange peel effect on the protector... which made my ipad look that much messier with my fingerprints.
After researching for awhile hoping for a better product to protect my ipad, I came across someone talking about a brand name Skinpea. I took the chance and purchased from Skinpea... It made a BIG difference.
First, Skinpea's screen protector was ANTI RIPPLY with NO ORANGE PEEL.
Secondly, the screen was very smooth and clean. I can already tell by touching it that the quality was very well made and felt safe putting it on my brand new ipad.
Ofcourse, you can't get rid of your fingerprints 100%, but i can tell you that the fingerprint issue got better than not having the protector.
I'm Skinpea all the way now, I got it for my phone and my mothers new phone.
Hope it helped.