It's pretty typical to run cat 5 for both voice and data these days. The downside in your case looks to be that it seems they only an one cable to each room. That means that you could only have one, phone or data, in each room. The left side is voice and the right is a patch panel for data.
You need a tool like this to punch down cables in the panel
http://www.memoryexpress.com/Products/MX3172
And get some cat 5 RJ45 ends to replace in the rooms
http://www.memoryexpress.com/Products/MX244
Wiring sequence for cat 5, you want to wire 568B
http://crimsonshift.com/make-cat-5-cable/