One of my favorite routes is going up the Ammonoosuc Ravine Trail and down the Jewell. Ammonoosuc is quite steep for a time, but if you take it slow and allow plenty of extra time, you'll enjoy it, and it gets you to Lakes Of the Clouds Hut which is a good lunch stop before you make the push to the summit. The Summit from Lakes of the Clouds does not have much elevation gain, which is nice. Descending via the Jewell Trail is nice because it's gentle at about the same time of day when you really need a break. It also gives really nice views of the other side where you hiked up, a sense of accomplishment.
The Auto Road seems like an easy option for walking but it will keep you taking the same kind of steps, using the same muscles, all the way down.... I've done it and several hiker friends of mine have done it - ONCE and never again. Trails are a bit easier on your legs because they work different muscle groups. One minute you're sliding past a boulder, the next minute you're tip-toeing over a muddy section... it changes. Walking the Auto Road (down) was "clop, clop, clop, clop, clop, clop, clop..." for 2 hours.
If you hike Mount Washington, DON'T wear cotton. Believe it or not, "hypothermia weather" is actually +30 to +50 F, and damp... exactly what you'll probably find in the Whites in summer. Most of the time when people get hypothermia in the Whites in summer it's because they were wearing the same cotton clothing they'd wear around town in the fall. Cotton holds moisture close to your skin where it chills you. Get the wool, the Poly-pro, etc. because they wick moisture away from you and they dry quickly, and bring layers you can add as you go up. Also bring high-protien, high-calorie foods and 2 quarts of water per person.
The AMC has a list of hike preparedness on
www.outdoors.org. You can also call the US Forest Service in Conway and their info desk will help you.