I used to hate waking up. Now, it's one of my favorite times of day because it's a fresh, private time of day. It can be an entire practice in itself without having to feel self-conscious when everyone else is awake.
When I wake up, I think of a gatha I read by Thich Nhat Hanh: "Waking up, I smile. 24 brand-new hours are before me. I vow to live fully in each moment and look at all beings with eyes of compassion." It helps me really feel the preciousness of this human existence and remember compassion from the very start. Then I take three deep breaths, expelling drowsiness and last night's dreams on the stale breath. After, I do five to ten minutes of Green Tara visualization in bed.
I try to stay 100% mindful of my morning routine of showering and getting dressed. There are lots of interesting sensations you experience at no other time of the day, so it's a great opportunity for mindfulness. You can also do some poignant reflection on the human body. After all, when else are you naked on a regular basis where you can see/touch your weird human body?
I wear some prayer beads around my wrist as a reminder of the dharma and compassion. Before I put them on after a shower, I recite "om ah hum" seven times, blow on them three times, and recite an "om mani padme hum" for each bead before putting them on. There are only 20 of them so I don't hog the bathroom forever!
I don't know if this is the "right" way but I find it meaningful. But all this does help me start the day with thoughts of mindfulness, compassion, and the dharma. Sure beats being grumpy and barely awake.