It’s entirely your decision, but having a screen protector will reduce the chances of you accidently scratching or damaging the screen. The Desire is my first touch-screen phone too, but I’ve always used screen protectors in the past - if nothing else, knowing that I can peel off the old protector and apply a new one and keep the screen underneath looking as good as new provides some level of reassurance.
A good screen protector will be almost invisible and should not affect the display quality noticeably at all (an anti-glare one might diffuse the image slightly, but that's the price you pay for reducing reflection). Similarly, there should be no sticky residue when removing them.
I’ve always used Martin Fields screen protectors in the past, but this time around I went for a Brando ultra-clear one and I’m very happy with it. The only slightly tricky bit is avoiding dust and dirt getting in there when sticking the protector on in the first place!
Of course, this is also the first phone I’ve ever had with an exposed glass screen, so perhaps I’m being paranoid as it's probably more resistant to scratches than a plastic one - I’d rather not take the risk though.