I had an idea, but wasn't quite sure enough of my astronomical intuition to give it without support. Fortunately, I found several places that gave clear descriptions of what it would be like to be on the surface of a ringed planet. Since Saturn is the most well-known ringed planet, I focused on it. (Of course you can't stand on the surface of a gas giant, so these descriptions are hypothetical, but still quite helpful in answering your question.)
I'd like to add that I'm assuming you're talking about highly-reflective rings made mainly of ice particles, like Saturn; the darker, dust-based rings surrounding the other three gas giants in our solar system are very hard to see, and would make much less of a difference in what we'd see in our sky.
With that said, here are a few descriptions that I found:
The rings would be dazzling sheets of light arcing across the dark Saturnian sky. Their brilliance would be most impressive just before dawn or after dusk, when the Sun would be below the horizon, but the illuminated parts of the rings would be spectacular curtains hanging above you in the star-filled sky. ("Saturn: The Ringed Planet")
Although Saturn has no solid surface to stand on, humans may someday view its rings from close range. They may walk on some of its icy moons or even float above Saturn's clouds in big balloons. From such a lofty vantage point, the rings would form wide, sparkling bands across the sky. Sometimes, icy particles from the inner edge of Saturn's rings may fall into the planet's atmosphere, creating bright "shooting stars" as they streak through the sky of this delicate giant. ("Saturn")
[I]f you could [stand on Saturn], they [the rings] would look a little bit like the Milky Way looks to us: a stripe of light in the sky. The stripe would be very narrow if you were standing right below the ring, and it would look wider if you were standing towards the poles of the planet, away from the ring. ("Surface of a ringed planet")
And for an artist's conception of what the rings might look like from the surface, see this site
As several people pointed out in an astronomy forum thread
, rings would make astronomy somewhat more difficult by lighting up the night sky and obscuring our view of fainter objects. Even from the winter side of the rings (with the sun shining on the opposite side of them), they'd be visible; for some absolutely gorgeous pictures of Saturn and its rings unlike most you've probably seen, check out these pictures from Cassini
, where the Sun is behind Saturn, leaving only the backlit rings. Seriously, they're stunning.
It would be awesome to have rings like Saturn's.