Are you driving or flying?
If flying, you should consider acclimating to the altitude - nearly 8000 feet.
If driving, you will have a little time to gradually acclimate.
Drink plenty of fluids - avoid alcohol and caffeine.
Consider taking one aspirin (real aspirin) per day.
Climate - yeah I ;ve seen it snow of the Fourth of July -
But you'll more likely have chilly weather - esp. in the mornings.
You should expect highs in the high 60s to lower 70s; lows in the 30s
Roads - the official YNP website has a road status webpage.
Sylvan Pass is undergoing a complete reconstruction - avoid.
Other than that - most roads will be O.K. I am miffed at the NPS
for providing lots of pavement for RVs but none for syslists.
Most Yellowstone roads are narrow with no shoulders and heavy traffic.
You can make lemonade out of lemons by riding early and late -
Then using the middle part of the day for hiking and exploring.
Nearly every campground has hiker/biker campsites so you ride pull in late.
Early morning and evening are also the best times to see wildlife.
A few roads have shoulders:
West Yellowstone to Madison
Madison to Old Faithful
West Thumb to Lake
Gull Point Road is short, but very nice. Also you can ride from
Artist Point Road to the Canyon Loop on the old roadbed.
Dunraven Pass Road has been recently rebuilt -
But it is still steep and narrow with impatient drivers.
Try to avoid the huge campgrounds - or hit them mid-day for showers.
The smaller campgrounds are much nicer for quality of experience.
These include: Lewis Lake, Canyon (great hiker biker), Norris (!!!)
Indian Creek (great trail), Tower, Pebble Creek
You can bike into one backcountry campsite - Lone Star.
You'll near a permit and to store your food in bearproof containers.
If you are flying in to Jackson or driving -
I would HIGHLY recommend a day in Grand Teton.
It is more cycle friendly - stunning views -
considerably lower to altitude adjustment.
Hiker/biker campsites right on Jenny Lake!!
Have fun - J