Safari it not very compatible some websites on internet. I'd recommend to download and install Chrome. Chrome is a similar browser like Safari and Chrome is from Google. It is compatible with all websites out there
Download and install Chrome from here --> www.google.com/chrome/browser/desktop/
After installing Chrome, open it. Then access your online accounts in Chrome.
You can upgrade your OS X using App Store application. Open App Store application from dock or from Applications folder (In Finder, click on Go menu at the top and select Applications. This will take you to the Applications folder).
Open App Store application. In App Store, you will see the latest OS X (latest is 10.11 El Capitan). Click Get or Download to download it. Once it is downloaded, it will automatically asks you to install it. So do it.
If you are already on latest OS X, then click on Updates tab at top in App Store and do all available updates. Still, I'd recommend to download and use Chrome as Chrome is compatible with all websites. Let me know how it goes please.
If you want I can remotely connect to your Mac and do all these steps for you while you watch. This is only an optional additional service. Let me know if you want remote support.