Internet connection "limied", no access to some sites,apps

Power off the router. Unplug it from the wall. Wait a while.

Plug it back to the wall. Power the router on. Wait until all the lights are lit properly. It will take a while.

Restart the computer.

For more on this:

http://support.apple.com/kb/HT4628?viewlocale=en_US&locale=en_US

Back up all data.

From the menu bar, select

 ▹ System Preferences ▹ Network

If the preference pane is locked, click the lock icon in the lower left corner and enter your password to unlock it. Then click the Advanced button and select the Proxies tab. If any proxy options are selected, make a note of them and then remove them. You don’t need to change the bypass or FTP settings. Click OK and then Apply. Test. The result may be that you can't connect to any web server. Restore the previous settings if that happens.

Do you have the same trouble with all wireless networks, or only one? Are there any other devices on the same wireless network, and are they having trouble connecting?

Have you tested it on any other networks? For example, can any of your phones be used for tethering?

Check that your router is set up according to these guidelines:

iOS and OS X: Recommended settings for Wi-Fi routers and access points

Boot into Recovery mode. From the OS X Utilities screen, select Get Help Online. A clean copy of Safari will launch. No plugins, such as Flash, will be available. While in Recovery, you'll have no access to your saved bookmarks or passwords, so make a note of those before you begin, if they're needed for the test.

Test. After testing, reboot as usual and post the results.

If you had the same problems in Recovery mode, then nothing you can change in the configuration of your computer is going to have an effect. The cause of the problem is in the router or the network. I doubt that interference has anything to do with it, because that would affect all websites equally. That's about all I can tell you.