Mail it to me and I will take care of it for you ;)
Kidding, of course. Here's what I recommend:
The single best thing you can do for your machine is get a SATA SSD into it. That will 10x the performance of your system for most tasks immediately. After that, max out your RAM at 16GB DDR3.
Assuming a 500GB SSD, you can do both of these things using new parts for less than $100, and if you get used parts, for less than $50.
A system with a 2nd-gen i5 or especially i7, 16GB RAM, and 500GB SSD storage will be fast enoug for essentially every modern computing task that isn't modern gaming, graphic design, video editing, or complex programming (it'll be good enough for simple coding tasks). You can do whatever else you want.
As far as batteries, my recommendations are twofold: Get the highest-wattage charger you can (probably a 170watt for the T520, I think) and the largest battery possible. New ones are available on eBay. You want the "extended life" models. They're not great, but they should get you a few hours of usage. Depending on what you get and where it's from, it's another $20-$50.
