If I am capturing leads from my marketing activity, how do I nurture them, keep them live, and move them towards close?
You want to do different things depending on the type of lead. Examples:
Nurturing leads is all about the relationship. Jason has some excellent tactical points on the mechanics of it. What I have seen work is that you make your company a trusted source of valuable information. Think of it like a colleague that you keep in touch with or ask a question about specific topics. Doing this creates a mutual benefit -- your customers get what they need and you potentially get a sale. This is a long term, strategically focused approach.
Check out this excellent guide to lead nurturing on the Marketo website.
(I'm not affiliated in any way, just wanted to point out a FANTASTIC document.)
Interested in any other answers and discussion from the assembled...
Surprised the obvious answer hasn't been posted - buy Dharmesh's book :-) - it's got a couple of chapters around this area, one specifically about nurturing leads.
Create a sales system. The others are more quailfied and have given great suggestions that will save you a lot of time. In order to find out what is working, you need to track your leads, marketing approaches, phone calls, email, tweets, chats, etc. As prospects move through your pipeline and either close or die off, you'll be able analyze this process.