This 100% depends on your space, the direction your back yard faces, and how much money you have. Post a pic and you'll get more help.

We live in MN which is probably colder, snowier, and buggier. Our last house we did a patio. It was beautiful but we couldn't use it much as we got eaten alive (it was shady and on the north side of the house). This house we have a walkout basement and built a deck. We've been able to enjoy it a lot this summer - seems the breezes here are better here at keeping away the mosquitoes and the elevation helps as well. Also our southeast facing back keeps it sunny most of the day, but the sun moves to the front after about 5 PM so it's shaded in the early evening. No concern with shoveling (my parents never shoveled their deck either - they should be designed for snow load), and using composite decking takes care of the resealing aspect.

If we could have afforded it we would have done a screened porch but quotes for that were over $80k a year ago which was a bit out of our budget. If you can afford that, a screen porch would probably be your best option. Adding a gas fireplace to it will extend your use of it into the fall and earlier in the spring. But also $$$$.

Thanks. The back of our house faces south. That is where we currently have our patio table, but it does need shade as the setting sun blasts in from the west at dinner time. The east side of our house is shady in the evening, and also would look good with a covered porch. Our deck in Maine starting sagging pretty bad from snow load so it depends on the quality of the construction I guess.

A roof won't help with the setting sun - for that you might want to look at shade trees or a wall of arborvitae or something similar. If an umbrella doesn't help, a roof won't either.

With a lot of bug problems, a screened porch is the best solution.

I am a New Englander also, I would look into a screened porch with screens that can be changed to windows. We have a deck on the south side of our house with an awning but at dinner time the west setting sun is right in your eyes. The original deck was mahogany which was supposed to last forever but did not so after 25 years we replaced it with trex. It is a cape style home and I insisted on replacing the wood gutters with wood gutters but eventually they failed. So as I get older all my purist ideas are being replaced with low maintenance materials

We like our 3 season room. 5 ft windows on 3 sides. In the winter I call it my walk in cooler. I use it at large holiday gatherings to hold some foods instead of jamming up the fridge. I also store my Christmas cookies in it. It does have electric baseboard heat , but we only use it when we need the extra space for company during winter. It would be too expensive to run all the time.