The benefit to moving the whole thing outside is you could close it up at night to retain some heat if we get some cool nights. If the night temps won't go much below 50, I wouldn't worry about the greenhouse.

I wouldn't put the greenhouse or the plants immediately on the west side of house if it gets full sun. Start with morning sun, afternoon shade so they don't burn. The greenhouse would also get extremely hot if not ventilated.

Thanks ginkgonut, So I would probably want to keep it open all day. And for now close up at night? My wave petunias still have alot of growing to do they are about 6 weeks old. Would it be better to keep them in for now under lights? And is putting the other plants outside in the greenhouse for the next week or so good enough for hardening off?

It's warm enough to harden them off outside now without a greenhouse. The key thing is not to expose them to full sun for more than an hour or so the first few days. They will cook, much like people sunburn if they hit the beach for hours after no full sun exposure. Baby them, as it sounds like you have. Gradual exposure is key. These cloudy days may be a pain to us gardeners, but it's actually ideal for acclimating plants to the outdoors. There is also less transplant shock to the plants when we place them in the garden with cooler temps and not a lot of sun. Hope you have much success!