Trimming your dogs nails is the most stressful job in maintaining your dogs health. Unfortunately it's also necessary. Your stress is no secret to your dog who will join you, thus making the task harder than it has to be.
Here's a video from Dr. Becker demonstrating how to trim a dogs nails and how to make it as stress free as possible:
Having caused a few bleeders over the years I can't stress how important it is to have styptic powder at the ready. The amount of blood that can come out of a nail cut too short is alarming so be prepared. Use the powder and pressure to stop the bleeding.
Once the bleeding has stopped I apply a drop of Krazy Glue and allow it to dry before releasing the dog. I don't know how many vets would endorse this practice but it keeps the flow of blood from reoccurring.
Of course the other alternative is to let your vet do it. It's not expensive and some vets will even do it for free for regular customers.