Updating my old macbook to new OS

Since yesterday I've been trying to get my Macbook a new life after it became very slow through the years. I've done a internet recovery and installed the OS again. The result is a faster Macbook, but the OS, apps, and browsers a very old. And I'm not able to get these things updated.

The system says there are no updates. I found information on internet that I could download a file on Blukap.com, so that I can update from the current OS to the newest OS via the OS's in between. But when I go to the website of Blukap it says my browser is no longer supported. When I want to go update my browser, it says 'the computer will no longer receive updates because MAC OS x 10.6 - 10.10 are no longer supported. So I am in a vicious circle, where I can not do anything to get my macbook to current standards anymore. Is there a solution?

Current OS is version 10.7.5.