There are several entities here. The NH Parks own the top and lease to the Mount Washington Observatory, a non-profit. AMC does the trails. The Cog and the Auto Road are separate commercial enterprises.
Here's the quickest way to the top:
http://www.youtube.com/watch?v=s4TmKodpqx8
Other trips. I've done the cog a couple of times and it's worth it just for the experience. There used to be a discount if you stayed at the Mount Washington Hotel. Maybe still? Technology has changed, and the first and last cog trip of the day are steam. Other trips are diesel and a little quicker. The Observatory has been working with the Cog to give you more time at the summit. The hotel is a great experience. I've also done the auto road a couple of times. Went up once in the winter in the sno-cat and spent the day with the observatory crew. (I won a TV auction prize.) I've also been a member of the MW Observatory for several years. Along with some great off mountain benefits, this gets you a tour that is not open to the public. They have some extraordinary mountain top once in a life time events that are not highly publicized, but are highly memorable.
If you're not big on fancy hotels, stay at Joe Dodge Lodge, but first join the AMC for big discounts. My mother-in-law, when she was 75, hiked up with a group of similar age active friends. A two day hike with an overnight at Lake of the Clouds. Yes, she is active, rows and kayaks along with hiking.
Last time we drove up with two cars, (June 1) went into the clouds at 4,000 feet, although three days before and two days after were beautiful and clear. When we got closer to the top, visibility was only 25 feet. On the good side, we had the mountain top to ourselves. On the way down, the lead car stopped at 4000 feet to take beautiful pictures of the valley. Driver and passenger got out and the passenger locked and closed the doors, engine running. Oops. Fortunately the owners of the Auto Road patrol about every 15 minutes (they prefer to not have AAA running up and down the road) and they were able to unlock the doors quickly.
In sum, it can be a very easy day or a difficult day, so be prepared for an adventure, and be prepared to prevent an adventure. No matter what, you'll have some great NH memories.