Yes, do light exercises like wiggle yourself like a worm in bed when you wake up. Do other light exercises in bed to get rid of your stiffness first to improve your blood circulation before you do your stretches.
When you sleep, your body is either in one position or not moving much which causes the stiffness and blocks the blood circulation. So move your head, neck, arms, legs, body, fingers, toes left and right slowly to get the blood flowing again.

I head straight to the shower and warm my body up. Have a half banana or something, take my anti-inflamitory and muscle relaxer, wait 30 minutes or so, then do slow, easy stretches. All though not every day as I should.

This is what they call "boom and bust"

I do bicep curls without weights for example because my arms have been weak and painful.
Think 10, do 5, and work up to 10 repetitions. Don't push through and do 20
Little and often is much better.
You need to increase blood circulation and warm the body up
I would not do stretches on a cold stiff body.

The suggestion about the shower is good! I also rub some kind of meds on the areas I know will hurt before I stretch. The more you stretch the easier it gets.

I have gone to PT twice a week for the past year after complications from a hysterectomy. My PT suggests that if it hurts dont do it. Try to modify the stretch so I dont have increased pain. Start very slow and as your body accepts the stretch try to stretch a bit more. She stated that we are already in so much pain who wants more pain. However, we won't know if the stretch is too much until we actually start stretching.