Best route for Alpine Garden depends on whether or not you just want to see the flowers or if you need to spend a few hours up there, doing photography.
If I'm going just for a look-see, I'll go up the Tuckerman Ravine Trail because as soon as I get out of the ravine, I've got the Alpine Garden Trail right there on my right. However, I believe that this trail is still closed through the ravine, so the Lions Head Trail would be the way to go -- like the Tuckerman Ravine Trail, it will put you near the southern end of the Alpine Garden, too.
And...if I'm going up top and need the maximum amount of time for photography (several hours), I'll forego foot power and take (yes), the Auto Road and park at about mile 6.5, where the Huntington Ravine Trail terminates next to the road. There's a parking area for about three cars on one side of the road, and another for a couple of cars on the opposite side. From there, it's 0.3 miles to the northern terminus of the Alpine Garden trail.
Hope that this helps...