Hey optix,
I think the best option for you would be to look around at some autoresponder/mailing list programs like you said (I found this one:
http://www.aweber.com). You wouldn't need to be connected all the time. All you would need is to sign up with them, and they'll give you some code that you could put in your website and it would show that form.
They would also let you specify what page people are taken to after they put in their email address and click the button.
The downside is that they aren't free. You'll have to spend a couple of bucks at it to make it work.
By the way do you have an autoresponder setup on your website? If you do, and you don't want to buy a mailing list program (which you should buy if you can) then I can give you a quick script which sends an email after they submit their email address and takes them to a page that you specify.