Driving on the "other" side of the road is easier for some people than others.

You should also practice around Las Vegas before heading out on the road - but once you get rolling - you will hopefully be OK. And unless you are a truly skilled driver - get the automatic transmission - which is how most cars come equipped from rental companies in the US.

When I have been in the UK and other places that drive on "your" side of the road - I always tried to remember when starting out - to look BOTH ways very carefully - as traffic is coming differently than you are used to - and - always try to keep the center line next to your outside shoulder/driver side window - and once you are into the flow of traffic - it's gets easier.

From Vegas to Zion - is about 3 hours - and Bryce - two more hours - or longer - depending on how much you want to stop and enjoy the scenery between the two parks.

The Grand Canyon is 5 hours from Las Vegas - straight shot back. (Check the maps - as getting from Zion/Bryce to the GC takes a little round-abouting - assuming you are going to the more popular/larger - South Rim - vis a vis the North Rim)

Suggest you also stay on the "Rim" at the South side - perhaps Bright Angel Lodge. http://www.grandcanyonlodges.com/bright-angel-408.html

As for car rental companies - all the big names do a pretty good job - and here is a website I start with - www.carrentalcar.com,
and some of the "smaller' car companies I have used (such Fox) were fine too - but be sure to thoroughly check out your rental car before driving it off the lot.

Oops. That's www.carrentals.com Need the "s" to get the site that lists all the rates.

I think the only problem you would have would be in Vegas. There is a lot of traffic there, but once you get going on the main highways, it won't be that bad. Where I always have a problem, when driving in the other lane, is when making a turn and remembering to make sure and stay in the lane I am supposed to be in. Actually, I think in some ways the less experience you have the better. This way you aren't relying on your instincts as much. I'd say go for it!

We didn't have any problem driving on the left side of the road in Ireland - and they have very narrow lanes. Every time we got in the car in Ireland we would say, out loud, "To the left!"

We had to say "To the right!" a few times after we returned home.

Just pay attention. I also think that it will be less stressful driving once you leave Vegas - except you'll want to look at the scenery while driving. Don't feel obligated to drive faster just because Americans may be driving fast.

I haven't noticed a huge difference between car rental companies in Vegas.

One caution: When outside of Vegas, gas stations are few and far between. Top off your gas tank at every opportunity.

We did this trip last June with a family of four and it was wonderful. From Vegas stay at the best western near Zion then drove to north rim and stayed on site. There are several
walks that provide fantastic views. Next drove to Bryce and stayed in park we got there in time to dine at the nice restaurant in park. Next we drove to Zion hiked, swam in the river and went canyoning. Lots of great places to eat in town. We planned the trip this way because it was an easier drive back to Vegas on the last day. Enjoy