In regard to the battery purchase, there are so many places online, I can't really say one or another is a better place. I can say, however, that you should be careful of the battery purchased. Some after market ones just aren't worth the time and effort to get as they don't work correctly. Not all, but many. Me I stick with OEM batteries myself (for laptops at least).
On removing it well it depend son the laptop. Luckily, for many devices, you can find videos online walking you through changing out the battery. Just do a quick search for videos on your device (Make/Model).