In my opinion, the best light weight Linux distro that also performs well is called Porteus Linux. It is very fast and simple. The taskbar looks transparent and the overall look of this fine distro is very nice. It runs very fast on low powered processors like the A4. This distro also runs just fine on older computers.
It is wise to use Firefox on light-weight distros because Chrome is resource intensive. Unless you are using a Chromebook, chances are Google Chrome will use lots of CPU on just about any machine.
If you do want to install Porteus onto your Hard drive. You must first make the Live USB using Unetbootin after you download the ISO. After that you must mount your hard drive to dev/sda1 and then use the Porteus installer to select the hard drive installation option and not USB.