Cat 3 is ok, but cat 5 is better and the difference in price should be pretty small for your project.
The rest of your house probably has the old 4 wire cable anyway.
You will only use two of the wire pairs in the new cable and most of the cat 5 you will find has 4 pairs. The blue and orange pairs would be the correct ones for typical phone lines. The rest of the unused wires can be wrapped back around the cable at the ends to keep them out of the way. Your old cable probably has green, red, black, and yellow wires. With the new cable you attach the mostly blue to the red, the white with blue stripes to the green, the white with orange to the black, and the mostly orange to the yellow.
Get some kind of jack to install at the new location and run your wire. There are dozens of decent tutorials on the Internet about this topic so I won't go through it all step by step. It's not difficult however. Just take a look at how your existing jacks are wired and try to replicate that.
Make sure that you don't have any phone signal before you start. These are low-voltage lines but you can still get a decent shock if a call comes in while you are working.