The non-opinion-based, objective bit
Installing most grid systems on top of your website will likely break some aspects of it. You can’t really just plop one in and just use it when you need it, you need to start the design with it and carry it all the way through.
If you want to make it responsive without having to rewrite from scratch, look into media queries, because that’s the magic of it.
A CSS framework uses a lot of media queries for the responsive design bit, then bundles lots of stuff on top of it. If you just want the responsive side of things, you can implement your own media queries (common breakpoints are 0-640px for mobile, 641-1024px for tablet and 1025px+ for desktop.
The bit that can get opinion-based if you’re not careful (but hopefully isn’t, here)
There are a lot of popular frameworks for responsive design (and the whole mcguffins that goes along with them), with two of the most popular ones being Twitter Bootstrap and Zurb Foundation. I’m not promoting discussion of them here, because SO isn’t for opinions, I’m just saying that they exist and are widely considered “robust enough” for production work if you want to go look at them. There are many others which are also robust enough.