Okay, assuming you made it to the Ahwahnee parking lot, head north toward the cliff until you hit the hiking/biking/horse trail, and follow this eastbound for 3-400 feet. Watch for a stream-bed crossing, and follow this stream-bed up to the base of the cliff. In the springtime, this streambed will be full of water.
The first pitch is a right-leaning chimney, and you will find it just to the right of the streambed. In the springtime, this whole wall runs with water, and you will get wet if you climb then.
Pitch 1: Manky, ultra-slippery chimney. Roper has it at 5.5, Reid at 5.6. It's at least 5.6, just because it is so smooth.
Alternate pitch 1 suggestion from Misha
If you don't want to deal with the 1st pitch chimney for one or another reason, you can walk around the base to the right of the chimney for 200-300' and climb a 5.8 crack/face that will get you onto the 3rd class ledge above the classic 1st pitch. This is a great variation as it features clean rock with sustained 5.7 jamming and face, and occasional 5.8 sections, way more pleasant than the chimney (IMHO)!
Pitch 2: 3rd class scramble along ledges to a short section of 5.4 crack in a face.
Pitch 3: 3rd class scramble along ledges to a short section of 5.6 crack climbing on a face. Don't go in the chimney on the left. It's yucky.
Pitch 4: 2nd-3rd class scramble along good ledges/terraces, and then a short, strenuous (for us soft climbers) bit of 5.7 jam crack. Son't go astray here, the ledge continues upward for a short while past the crack, but turn left at the crack (you can't miss it - the jams are piton scars, and the rock is polished by the thousands of climbers that clamber up each season).
Pitch 5: Cracks and flakes, easy going at 5.5, directly up the face.
Pitch 6: More cracks and flakes, easier yet at 5.4.
Pitch 7: 5.6 flake system with good pro. Short pitch.
Pitch 8: Stay on the left side of a small tower/buttress. It looks easier to go to the right, and initially it is, but soon becomes more difficult.
Pitch 9: The famous pendulum pitch. A short traverse at 5.9+ (McNamara of Supertopo has it at 5.10b, and I think this is more accurate) or a little pendulum on fixed gear. If you are climbing in the early springtime, this pitch will be running with ice-cold water, sometimes the cascade is 20-30 feet wide, but usually not very deep.
Pitch 10: The 5.7 bear-hug flake. I've seen it laybacked, but for me, gripping both edges of the flake is more secure. Move up this flake until you can grab the tree branches, and then the trunk.
Pitch 11: This pitch makes it all worth it. This is where the Rotten Log used to be, a piece of old tree that fell down and lodged, allowing easy crossing of the steepest part of the route. The log is now gone, and in its place is a steep little headwall with a perfect finger crack at just less than vertical, with good pro, and with immense exposure. The climbing on this pitch is scary but relatively easy.
Pitch 12: Parallel cracks are the highlight of this fun pitch, coming in at a casual 5.6. Follow these to a large Jeffrey Pine adorned with multiple slings.
Pitch 13: A little face climbing, weighing in at 5.5 gives way to 3rd class scrambling.
Pitch 14: Exposed slab climbing to the rappel/belay anchors.
Pitch 15: If you rapped, then you missed this pitch. It's a very cool pitch, 5.4 friction along the crest of the face with 1,400' of air beneath you. To make it sporty, it is often covered with sand or pine needles, or maybe oak leaves. Once you gain the trees, follow the use trail up to the rim and double back toward Washington Column.
Comfortable shoes, standard (light) trad rack, stoppers and a few cams, lots of slings, 24" & 36" are most useful. All the belays have stances. If you're not doing the rappel, bring some walking shoes.
You have a couple of options here, both of which I've tried. At the end of pitch 14, there is a 2-bolt belay/rappel station. The first rap is the killer, you have to back over the edge with 1,600 feet of air beneath you. Going over the edge at night is easier, you can't see how far it is to the ground, but in the daylight makes it much easier to find the next station. You need 2 50-meter ropes.
There are 9 rap stations, and they tend to the rapeller's left. The 3rd or 4th rap goes off an overhang, so it is free-hanging for a bit, but most of the raps are slightly less than vert, making for some tough rope drag.
If it is raining, rapping is a bad option, as this part of the face carries a lot of water, and if you're rapping, you're getting soaked. A few of the stations are ledgeless, just 2-bolt stances, and one station is a scrawny little tree growing out of a crack. Don't study it too closely, or pull too hard!
Peter Carrick and friends re-bolted the rappel stations in October of 2000, so all of the stations are supposed to be bomber now. This excerpt from the ASCA page: "Replaced 8 bolts and replaced 15 hangers with Fixe double-ring rappel hangers. No more ugly webbing, but that also means the stations are not as obvious."
The second option is to hike over and descend North Dome Gully. The hike takes a few minutes as you have to skirt the Death Slabs to gain the gully. Follow the use trail at the top of the climb, head east behind Washington Column, and stay on the trail. It will seem that you have gone too far, but there is an easy entrance to the gully. Don't rap! If you're considering rapping in, just walk a few more minutes and you'll find the entrance. This is the preferred descent.