Home-made Giant batteries
This is not my field, but as an outsider looking in on this one, I have a few thoughts:
1. I consider myself a practical engineer, but I enjoy pondering problems such as these. My pondering has turned to stark fear in this case because of the forces involved. Many people have never lived to explain what there experiment was about while working with nature's forces. I can only implore you to use EXTREME caution and thoroughly research anything you do before trying something. I'd try and find an experiment that has already been done and replicate it. This would at least give you some experience in the area before striking out and doing something that you haven't heard of before. PLEASE BE CAREFUL. You sound like you have a good, inquisitive mind and I'd hate to have that light put out before it could really shine - good minds are becoming scarce in the US these days...
2. In light of the fact that there are literally millions of buildings conducting electricity to the ground during storms I ask: why hasn't anyone tried to harness it before? My guess is many have and they found it to be unfeasible/impractical for one reason or another. I'd do much more research - I'll bet you'll find many experiments along the lines of what you're trying to do.
3. As for your professor saying "the wire will burn up", lightning conductors are on the tops of buildings worldwide and provide many years (sometimes centuries) of service without burning up. These are routed to the ground and usually connected to a metal stake driven a few feet down.
4. I remember that a few years back, National Geographic ran a story on lightning. In addition to some great photographs (as usual) the story had details of a kid at football practice that had his uniform blown off of him, and of the glass "roots" made by melting soil as the lightning is absorbed into the ground. Like I alluded to in #1, this is serious business...
My two cents...