Everybody's first rockclimbing trip to the Enchantments has to include Prusik Peak – that's a given. Now, what else to climb? After Prusik, you cannot, must not leave the Enchantments without exploring The Boxtop. This little chisel point on Temple Ridge packs so much variety and deviation to the point of perversity into a few pitches that it must be climbed to be believed. The standard route goes at 5.8. Allow at least a half-day from camp in the area. I've climbed it perhaps 4 or 5 times and never had to share it with another party.
From the W end of Leavenworth, WA, turn S on the Icicle River Road. Follow it through open valley, then W into the canyon to a very large trailhead parking lot at about 4 miles. Hike down the Snow Creek trail to the sturdy bridge across the Icicle River, then up and more up. Pass Snow Creek Wall (across the valley) at 2 miles, Nada Lake (5,000') at 5.5 miles, the little rock dam separating upper and lower Snow Lakes at 7 miles. Walk across the dam and continue around the left side of the upper lake, cross a bridge and switchback to Lake Vivianne, the first of the Enchantment Lakes, at 10 miles and 6,800 feet. The main trail crosses the outlet stream and continues on rocks around the S side of the lake, but you'll probably want to go up the hill to the right, instead, to camp near Temple Lake (aka Naiad Lake) or little Gnome Tarn, S of Prusik Pass. Find a viewpoint to scope out the ridge between Prusik Peak on the left, and The Temple on the right. Immediately next to Prusik is the very pointy Flake Tower, then the perfectly square-cut Monument. After a wider gap is the granite wall that is The Boxtop. Head for the base of the gully under the left end of it. This Photo from Paul Klenke is a good overview of the ridge. The Boxtop is the very square wall with "shoulders" on the right. Little Boxtop is the lump under the left shoulder. Beckey's guidebook recommends starting from the ridgecrest W of Little Boxtop and traversing under under the S face of that, but starting straight up the gully/chimney between Little Boxtop and the main peak makes an enjoyable warmup pitch.
The Enchantments are Red Tape Heaven. The area was so popular the Forest Service decided they needed to add concrete steps to the trail in places where it crosses smooth slab or climbs boulders. In other places they laid primacord or something across the rock to blast out little chips in huge smooth granite slab. Then – surprise! it was too crowded, so they established a permit and quota system for the area for camping anytime between June 15 & October 15. Not just a permit, reservations. Months in advance. Paid for in advance. For more details on applications & such, follow links from the Prusik Peak or Enchantment Peak pages.
Summer. Most years it's dry enough to climb by Memorial Day weekend, before the permit season starts. Or you might sneak up on a good October weekend, but if you can snag a permit and spend a few days in the Summer, you'll not regret it.
There are excellent campsites in the basin, and the quota system means you'll get a good one. It's all in the Alpine Lakes Wilderness Area, so you'll need a permit, as mentioned under Red Tape. No fires are allowed. Pack out your stuff.
The elevation of the plateau and its location E of the Cascade Crest just about balance each other out. The result is that the chance of sunshine on any given day is about the same as Seattle's on that same day. Of course the chance of getting snowed on is a little higher in the Enchantments than in Pioneer Square. The National Weather Service in Spokane, WA watches the weather on the E side of the Cascades. Wenatchee National Forest can give you trail updates.