DNS should never be used as a "failover" solution, as it does take a fair bit of time for things to propagate, and most ISP's that people use to connect to the Internet don't follow proper "time-to-live" rules anyway, so even if you DO change the values, it could take days for your end-user to see the change.
As well, the way you have it described, you'd basically split your traffic between the two sites, regardless if they were operational or not.
And, you'd have to deal with maintaining duplicate web sites on each server. A real PITA.
You also don't mention what kind of "stuff" you need on your web site? Does it use PHP, JSP, or another language to create the pages? Does it use a database? Are you using any kind of "built-in" software that comes with your current provider? (gallery, chat, etc) How much space do you need to host your files? There are a LOT of things that could impact on where you go for your hosting.
I've had really, really good luck using RackSpace for some of my clients (
http://www.rackspace.com).
You might want to give them a shout and see what they offer. (They offer a 100% network uptime, and 1 hour hardware replacement).
Odds are, though, that they're going to be too much for what you really need, and in that case I'd recommend checking out Mosso (
http://www.mosso.com) , one of their partners.
I've heard good things about them.
...jeff