I'm a huge fan of 120Hz monitors, with four of the awesome monitors in my lab. Onto your question - yes, I think you should upgrade to a 120Hz monitor, but you won't get the full benefits of the monitor in most games. The monitor, overall, will be a huge jump from a 60Hz panel.
This is noticeable everywhere, from sliding windows around on your desktop, or moving your mouse cursor - the fluidity of a 120Hz monitor is unmatched by anything else on the market. Even those $1500+ huge 30-inch IPS-based monitors with 2560x1600 as their resolution.
Of course, in games, it helps even more - but you have to be achieving a minimum of 120 frames per second before any of the benefits are met to 100%. But, just because you can't reach 120 fps doesn't mean you won't tell a difference. I find even at 60-80 fps on my 120Hz monitor, it is smoother than a 60Hz panel.
So I would suggest buying a 120Hz monitor, specifically Samsung's S27A950D - I run this and I love it. You can always slot in a new GPU in the future, but for now you can just turn down some of the details to achieve 120 fps. In the games that you can't crank to 1080p and achieve 120 fps, you can reduce the resolution to something like 1280x720 and keep the details at medium-high to achieve high frame rates.
Watch this space, too, as we'll be bringing you some 120Hz content in the very near future.
Chris has commented saying that the Samsung S27A950D has been discontinued, which is weird as I purchased my nephew one just last week. So I'll change my recommendation to ASUS' VG278H monitor - another 27-inch, 120Hz-capable screen. Newegg sell this bad boy for $599.99.