One GriGri, one Reverso
I can't think of a multipitch rock excursion on which I would not prefer this system. Belay the leader with the Grigri, bring up the second with the reverso in autoblock.
To rappel, either simul-rap (only on bomber, easy raps), or fix the rope for the first person (with the gri).
There are a ton of advantages to this system. The belay is always auto-locking, which adds safety and allows the belayer to take care of the rope, snap photos, eat and drink etc. Change-overs are cake.
When rapping on unknown terrain, having the leader rap on the grigri allows him to swing around, look for the next potential anchor, unsnag the rope, etc without worrying about holding onto the atc. And it's certainly much faster and easier than using a prussik autoblock backup.
With the new mini grigri, I of course like this system even more. Lighter (duh!) and it works with skinnier cords.
If you're worried about being out after dark, take one good headlamp (probably one with 3 AAA batteries and a good spotlight mode), and one smaller torch (probably one that runs on watch batteries, I like the petzl e-lite). You're partner can route-find with the powerful beam, and you can follow with the surprisingly bright little lamp. It's important to have fresh batteries, and it's always good to have spares if you're gonna be out for a few days. Lithium batteries are pricier, but lighter and stronger.
For most summer alpine rock climbs, you won't need a warm layer as the leader, since climbing will keep you warm. So bring one small puffy jacket for the team, have the second wear it at the belays, and bring a windbreaker for the leader. (It helps to climb with someone roughly the same size as yourself.) I put a lot of thought into clothing systems, so look for a future post with more details.
Kinda random, but sometimes you can coordinate footwear to save weight. Bring just one pair of running shoes, and have one team member run back to the base of the route to retrieve both climbers stuff. Or if the approach involves mellow snow and easy walking, bring one pair of runners and one pair of flip flops. The "leader" can kick good steps for the flopper.
One quick climbs up El Cap, I've brought one pair of approach shoes for the second/jugger, and one pair of flops for the other climber to amble down the East Ledges.
On single day pushes in the mountains, you are often required to "carry over" on a route, meaning you can't leave your pack at the base and retrieve it later. So, as always, weight takes center stage. Coordinate with you partner to see who has the lightest pack, maybe 30-40L for a day trip. Take that, and have the second person backpack-coil the rope(s) and wear their harness. For extra comfort, when approaching in a harness, don't wear the leg loops, but rather cinch up the waist and let the leg-loops hang off to the side.
On a two-day mission, you'll probably need two packs, but why take two big packs? Again, take one 30-40L main pack, and have the second carry a ultra-light 20l pack, one that stuffs down into nothing when not loaded, plus the ropes (backpack-coiled). When climbing, the second carries the big pack, and the leader has the option of leading with the small pack or stuffing it into the big pack.
BIVY GEAR, etc.
If your gonna be out for a few days, you can find a myriad of corner-cutting, weight saving tricks. For eating, bring just (you guessed it) ONE setup, and share. One small bowl/mug and spork are sufficient for eating re-hydrated meals, or you can even skip the bowl and eat out of the packet. Once done with the meal, cut the bottom of the meal pouch off and save it for use as a bowl or cup in the next meal.
For a sleeping kit, of course it all depends on expected conditions. Often when attempting a rock climbing objective, we rally during expected mild weather windows, so it is possible to go light. Just one night out? Consider sharing one sleeping bag, and just using it as a blanket for both climbers. Try and find a waterproof bivy sack with a full zip, so that you can unzip it to use flat as a ground cloth, or tarp if there's precip.
For longer journeys, I usually think that the benefit of warmer, more restful sleep justify the weight of two sleeping bags, but of course still look to use a light bag in conjunction with the other gear your carrying. If you're gonna bring a puffy belay jacket, look for a specialized bag with distributed insulation, warmer from the waist down, lighter from the waist up (where you'll be wearing the jacket anyways. If you're not gonna share a sleeping bag, look for one with a half-zip, or no zip.
Don't get locked in to what you MUST have on your harness. I've seen so many climbers obsessed with ALWAYS carrying cordalette, extra lockers, bulk webbing, knife, nut tool, whistle, etc etc etc. Of course sometimes as the second you might need slings in a self rescue scenario, but know how to rig everything with the slings and whatnot that you carry anyways. If the route is clean and well traveled, the leader doesn't need a nuttool, and you NEVER need rap rings unless you're trying to establish a popular rappel route.
More unsolicited advice to come, along with more pictures!