I had to do this to the family cat. It's the worst thing i've ever experienced in my life. We got her just as I turned 2, so i'd grown up with her all my life. She disappeared just as my parents went away and came back the next day, I had to take her to the vets who told me the only option.
I'm more than willing to say I was crying my eyes out as I did it. I walked out of the room and everybody looked at my funny until they realised the cat carrier that had something in when I went in there was now empty.
As people have said, remember that it's for the best, it's putting an end to their pain as well as yours. I can honestly say my cat seemed to know what was coming, and thanked me for doing it. She'd normally be a bit sketchy when it came to needles and things, but she didn't flinch, she just sat there and purred while i stroked her as the vet injected her. It's setting me off just typing this out...
Remember the good times, and although it's harsh to think about it now, a replacement pet can help make things easier, i know it certainly did with our family. It's not just a replacement pet to forget about the old one, it's a continuation of what you've been used to for the past 10 years.