Salvaging an old macbook

Hello all,

I have had my macbook since 2008 and for some reason never bothered to upgrade the operating system. However, now that Leopard is completely out of date, I cannot sync my phone to it (downloading the newest version of itunes requires a newer operating system).

My question is: how do I go about upgrading from Leopard to Mavericks? I'm pretty sure I have to upgrade to Snow Leopard first to obtain the mac app store, correct? and then upgrade from there to Mavericks, or is there another step in between?

Also, the battery is completely dead. I seem to remember the genius at the mac store telling me that I have a life-time warranty on the battery, but I do not have any paperwork to prove so. Will this be stored in my file at the apple store, or am I out of luck?

If not, I'm going to need to purchase a new battery and completely upgrade the operating system, which will probably require an external hard drive. Is all of this work worth it to recover this computer? I've never done anything like this before, so my knowledge on the subject is limited.

Thank you in advance!

Samantha