I'd second the notion of bringing a lot of water. The water sources are few and far between. It's a fantastic hike – highly recommended. I would not filter out of the Paria itself – it's full of silt and a lot of agricultural runoff. Also, make sure you are hyrdated once you get there – my skin dried out pretty fast. There isn't a lot of humidity in that part of the world. A good hat and sunglasses are recommended.
Get a book that discusses the route – one that provides maps of the seasonal water sources. We found it was very accurate and that good water was available.
Trail shoes are a bit of an issue. I hiked in cheapo trail runners that were thrashed by the end of the journey and completley water-logged. Some people take sandals, others take special-design water shoes.
We didn't see many snakes, but I also had a bad habit of reaching up and grabbing rocks without seeing what was on top. In retrospect, that was really unwise.
As far as gear went, we just took a tarptent and it was fine – in the dry climate it setup very easily and didn't suffer from the sylnylon sag. Some might argue that bringing a small weather radio would help, but we couldn't get ours to pick up anything in the canyon. You will have to check in with the ranger beforehand, and they will provide you with any weather-related updtates.
We went in late April, and it was unseasonably cool in the day (highs were in the 50s and low 60s) so it was nice to have a jacket along. I wore shorts the whole time. I brought a 20-degree down bag. I also went overkill and lined my backpack with a dry bag. I was paranoid about taking a spill (you are in water, crossing water, or just to the side of water about 90 percent of the time) and did't want wet stuff. I'd also completley recommend hiking poles….these saved me from a few falls when the river bottom suddenly became much deeper than expected.
We didn't have swim at all – but there were plenty of pictures of people who had to swim during especially heavy snow years. Check with the ranger for conditions prior to your departure. In that case, you will need to have something to keep your backpack and its contents dry. We read about people who actually brought intertubes with them so that there gear cold sit atop the tube as they pulled it behind them.
If you do a lot of photography, I'd suggest bringing along a lightweight tripod. A lot of the narrow canyons don't permit much light, and thus it become difficult to handhold without blur. Certianly, IS-enabled cameras set to ISO priority would be helpful in this regard.
It's a fantastic trip – one of my personal favorites. It will be a far cry from Santa Cruz or in my case, the Pacific Northwest. At the end of the trip do check out Lees Ferry – interesting story behind that!