My daughter and I started playing the first day, but servers have been down more than up for us. Lots of similar complaints and news stories about the server outages online. We were out late the fist night about 2am PST and were able to hit a lot of locations and gather pokeballs and items. We've got a fair number of pokemon too.
I was able to get on for a bit this morning and went to my first gym battle. I beat the two pokemon that were there, but lost server connection just as I won and it never recognized my win. I let the phone sit for 30 minutes before I shut it off and it never connected.
People are walking all over my town staring at their phones trying to log in. Most of them look disappointed. I overheard a couple walking in the water at the lake and one asked the other if they'd caught any water pokemon yet.