All my browsers will only open https: sites.

1. 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.

Start up in Safe Mode.

http://support.apple.com/kb/PH14204

2. Deselect Proxies if selected.

System Preference > Network > Advanced > Proxies Tab

Under "Select Protocol", uncheck any box if selected.

Click "OK" then "Apply”.

Try a restart.

Do a backup, using either Time Machine or a cloning program, to ensure files/data can be recovered. Two backups are better than one.

Try setting up another admin user account to see if the same problem continues. If Back-to-My Mac is selected in System Preferences, the Guest account will not work. The intent is to see if it is specific to one account or a system wide problem. This account can be deleted later.

Isolating an issue by using another user account

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 deselect 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.

Please read this whole message before doing anything.

This procedure is a diagnostic test. It’s unlikely to solve your problem. Don’t be disappointed when you find that nothing has changed after you complete it.

The purpose of the test is to determine whether the problem is caused by third-party software that loads automatically at startup or login, by a peripheral device, by a font conflict, or by corruption of the file system or of certain system caches.

Disconnect all wired peripherals except those needed for the test, and remove all aftermarket expansion cards, if applicable. Start up in safe mode and log in to the account with the problem.

Note: If FileVault is enabled in OS X 10.9 or earlier, or if a firmware password is set, or if the startup volume is a software RAID, you can’t do this. Ask for further instructions.

Safe mode is much slower to start up and run than normal, with limited graphics performance, and some things won’t work at all, including sound output and Wi-Fi on certain models. The next normal startup may also be somewhat slow.

The login screen appears even if you usually login automatically. You must know your login password in order to log in. If you’ve forgotten the password, you will need to reset it before you begin.

Test while in safe mode. Same problem?

After testing, restart as usual (not in safe mode) and verify that you still have the problem. Post the results of the test.

Start up in Recovery mode. In 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, restart as usual and post the results.

The cause of the problem is external to the computer. Restart your router and also your broadband device, if they're separate. If there's no change, the configuration of the router may need to be changed, or there may be a problem at your ISP.

Did you take the next step I suggested?