Q: Devices connect to Airport, but not to Internet

Hi all; here is my situation:

I just bought a new Airport Extreme to replace my old (Linksys) wireless router. In our house, we have two iPhones 6, as well as an iPad, Apple TV, and MacBook Pro that were all bought new this year. All of these things were able to connect to the internet via our old wireless router.

Since I set up and installed the Airport, various devices have been able to connect to it without actually being able to connect to the internet (i.e. on AirPort Utility, the AirPort connection is a green dot, while the actual internet connection is an orange one): first it was the MacBook having this issue while the phones and iPad were fully connected. After simply restarting everything multiple times- i.e. the modem, router, and MacBook- I finally got the MacBook fully connected (without actually changing any settings, which are all entered correctly). Now, however, it's the two iPhones that will connect to AirPort but not actually to the internet.

Any ideas? I have to say, these odd little idiopathic problems that always seem to pop up with Apple products are starting to really bother me... I still have never gotten my MacBook to connect to the Apple TV via AirPlay.