Hey everyone, I'm pretty much a complete n00b here, so pardon the... n00bness.
Anyway, I pulled out my dad's old road bike (It's a 1985 Corsaro Orion, looks kinda like this but w/ drop bars, no rack, and original shifters and brake handles). It wasn't used a whole lot.

I'm gonna fix it up both to earn some dough and just learn some bike maintenance. This is my first time doing anything like this. Other than obviously replacing the split tire and leaky tubes, and de/regreasing everything, what sorts of things am I going to need to check for/replace/adjust? I told the guy at the bike store about it, and he suggested replacing what he called the "deathgrip brake handles". How would I go about doing that?

Also, it's not very rusty, but how would you suggest getting rid of the rust that's there?

Thanks in advance for the advice and for not tearing me apart.