The road itself is not in terrible condition in my opinion, but for those who have never driven on a dirt road before it may seem intimidating. Be sure to have a spare tire with you, as well as a pump (12v pump ideally), and the knowledge to change it should you need to. The main concern with the road is the prevalence of slate rocks which can have quite sharp corners. Try not to roll over anything that looks like it has a knife edge and you'll get there just fine.
The road turns from pavement to dirt once you pass the visitor's center at the Ancient Bristlecone Pine Forest. From there, it is about 16 miles to the locked gate/trailhead, which will take about 45-60 minutes depending on the car you take and your comfort level. The first 12 miles of the dirt road is in pretty good shape, while the last 4 will require you to slow your pace a bit. The visitor's center is a good place to "take care of business," as they have a very well maintained set of pit toilets there. Aside - if you have a chance to visit the ancient bristlecones (oldest non-clonal organisms alive on the planet), it is well worth a stop either the day before your White Mtn hike to help with acclimatization, or as a fun side trip on your way out. There's a ~4.5 mile loop with minimal gain that floats around 10k ft elevation that I recommend.
The actual hiking trail up White Mtn peak is somewhat unspectacular, as it is mostly a 4wd road. It's rocky and barren almost the entire way, and exposed (the entire hike is above tree line) so bring a good wide brimmed hat, sunglasses, sunscreen, and CHAPSTICK with SPF. I forgot that last part and roasted my lips. Remember, risk of sunburn increases substantially at higher elevations. There is no water anywhere on the trail, so pack everything you'll need. I believe most people take 7-10 hours to complete the 15 mile round-trip hike. White was the first 14er for me and my buddy, and we did it in about 5.5 hrs moving time and 7 hrs total time.
Everyone has a slightly different opinion on best course of action for AMS, but in general if you can afford to spend 1-2 nights at elevation before the hike, it will help you. The more, the better. We stayed at Grandview Campground at 8.5k ft for one night, which is only about 1 hr from the trailhead. Nice sites, but no running water so take plenty with you.
I also recommend bringing either ibuprofen or acetaminophen (i.e. Tylenol) to treat AMS symptoms on the trail should you need it. Ibuprofen has been the standard for treating symptoms, but needs to be taken with food and can be a little tough on your gut while you're already exerting yourself. Tylenol is another good option without those side effects. Recent research (https://www.ncbi.nlm.nih.gov/pubmed/28479001
) found that Tylenol was about as effective as Ibuprofen for treatment of acute AMS symptoms. Of course staying hydrated prior to and during your trip will help as well. You may also consider consuming some form of caffeine prior to your hike, as it's diuretic properties work similarly to Diamox to alleviate some AMS symptoms. If you're not a coffee drinker, they make some pretty tasty Cliff Bloks with varying amount of caffeine in them. They make for great high-calorie high-carb trail fuel and come in a variety of flavors.
Well this was probably far more info than you wanted or needed but hopefully this will help some folks out there looking to bag their first 14er =) Good luck and have fun!