I'm going to be using an arduino as a computer of sorts on my bike. It'll be powering lights and a 16x2 lcd. I know this isn't a big power draw, but the reason I want to use a laptop battery is so that I can charge it once and not have to worry about it for months.
Here's the thing, I don't want to hook up a 14.4v 2.6 AHr battery to my arduino without some consultation. What would be a safe way to do this?
Through the barrel jack or into Vin (functionally identical).
Be aware that the linear regulator on the Arduino (or any linear regulator) wastes any input voltage over 5V. In other words, 9.4V (14.4V - 5V) is blown off as heat. To avoid this inefficiency you should use a device with a switching regulator; the guts of a cheap USB car charger would work or you can buy a switching regulator module off eBay (search for LM2596).
Module like this wouod work well also
Set the output to 7.5V and connect to the barrel jack.
I would have recommended 5V and connecting to 5V header, apparently Arduino's now have blown regulator issues doing that.