Hello!
I was pretty close to return my Venu being far from polite as it constantly refused to connect to my home wifi network (behaviour pattern was: the watch found wifi hotspots properly, but any attempt resulted always "no internet connection). Worth to mention, that rest of the devices at home (15+ iphones, iapds, notebooks, etc connect with no issues, only Garmin Venu says no). This problem seems to repeat in many questions/help requests here.
What I found is: carefully check your networks settings. I got two routers and a repeater at home. I tried everything, but as the last resort I checked DHCP settings and found out that too bad both "master" and "a slave" router had DHCP on. Normally, a slave one should have DHCP off to let only the master distribute IPs in local network. I reconfigured the network properly, disabled DHCP at slave router and....boom! Germin gently and quickly connected to access point.
Maybe that would be helpful for some struggling with connecting your Garmin to home wifi.
Good luck!