The key to having fun on the Royal Arches Routes is to avoid getting overly encumbered and knowing when to just unrope and walk. Lots of people epic here, usually because they move slowly on the easy first part. Note that this route can be crowded and that can affect your ability to move fast as well, though there are ample opportunites to pass, especially low down.
How much gear you take depends on your ability. The cracks vary in size and there is really no part of the route that is sustained, so there is no reason to go crazy with gear. For most parties except those that are pushing their limits, the less you take the happier you'll be. You'll often see people on this route going ropeless in just their sneakers and this makes for a fun outing. I've done it with a couple of different partners and we've always done the round trip in under two hours (the record is 52 minutes). In any case, this is a good route for each partner to carry a small Camelbak rather than a single pack for the second that gets switched at each belay.
There are many variations, but the classic start is up the polished chimney which involves a wee bit of struggle, especially with a pack, but the fall is safe (and unlikely) and you need to just get yourself up here, bring your partner up, and then unrope or short rope for a while (it looks like a hiking trail and pretty much is) until you get to the next step. The splitter crack on the face to the right is one of the pleasures of the climb, so don't stay left in the gully.
Do some more easy scrambling, either unroped or simul-climbing until you get to the next step. Again, stick to the crack on the face. When it looks like a hiking trail, coil the rope and walk to the obvious spot where the trail ends at the base of a 5.7 crack (see thumbnail photo).
Now the real climbing begins and it becomes harder to pass slow parties. Follow the obvious crack and corner system for three pitches until you get in the left-facing dihedral that leads you to the pendulum point. You can free this at 5.9+ (some say 5.10-) or you can just grab the fixed rope and swing over. There is no real reason not to try to lead this, as you can clip off high on the fixed rope and have pretty close to a top rope.
You now make a long traverse left which can be a raging stream in early spring. From the tree at the far end, you want to go straight up the splitter 5.7 crack and flake system, rather than fighting your way up the corner. When you reach the tree in the dihedral, squeeze past and traverse left to a ledge with a tree. Follow a nice crack up to a tree and then an easy traverse left on slabs. Follow the ramps to the top until you reach the rappel point. From here you can either rappel or make a quick jaunt across to the trees and walk down. Some people finde the "quick jaunt" terrifying actually, but it is in fact only 5.4 and low angle and I've done it several times in my running shoes and without hands.
If you opt for the rappels, you're on your own as I've never done it. From the top of the climb to the Ahwahnee takes me under an hour going down North Dome Gully (that's with some running; count on at least two hours if you have not done it before). I can't imagine I could rappel it that fast, so I've never bothered.
Assuming you're going to walk off... If it's early in the season and there's a lot of water coming down, you might want to stay roped up until you cross the slabs - they are often wet and slippery and a belay is not out of order. I regularly solo this route and, when these slabs are wet, I find it the scariest part of the whole outing. If they are dry, this is trivial. Either way, you climb up about 50' through the woods and then cut left until you can get up to the true rim, and then you wander up and right. The trail is usually obvious in daylight, even when crossing the slabs where thousands of climbers have worn off the lichen and left white "trails". When you cross the big slab below North Dome, at the end of the slab, you want to go up about 50 feet or so. You should never be bushwhacking. If you're bushwhacking, go back to where you lost the trail and try again.
Once you get to the top of Washington Column, skirt a big round boulder (inobvious, as the trail is blocked by bushes, but it is still there under your feet) and head down sandy slopes near the edge. Typically, when you have a choice, stay high. If you feel like you need to pull out a rope, you've lost the trail. This is long and merits a description of its own. Check out the one in the Supertopo Free Climbing guide as it is excellent.
Sorry for the blue cast to the photos. These were taken during a pre-work run up the Arches and I have never again taken a camera (the last three times up the route I did not see another person at all!). Someday I'll update these.