There are two main schools of thought: the "fast and light" / "light is right" / "rope, rack, shirt on your back" camp and the "be prepared" / "safety first" crowd. When taken to extremes, blindly following either ethos can lead to trouble. Like most everything in the mountains, it comes down to experience on similar routes, judgement, acceptable risk tolerances and even willingness to suffer.
However, these are some of the things I consider when racking up at the base of the climb:
Will you rappel straight back down the route when finished, or is there a long walk-off descent? If rappelling, then there is usually little need to take your approach shoes up the route with you. If there is a walk-off, then you will most likely need to take your shoes up the route with you (note that some walk-offs are not necessarily unreasonable to do barefoot, provided they are relatively short and have reasonably non-jagged / non-pokey terrain). Shoes are usually dealt with by simply clipping them off a rear gear loop on your harness---note that having lightweight, low top approach shoes or trail runners instead of high top hiking boots will make this much less annoying.
How long will the route take?
Neither routes nor climbing parties are created equal. Leading at your limit might take upwards of an hour for a single pitch, while a strong climber might lead the exact same pitch in fifteen minutes. More than almost any factor, the amount of time you spend off the ground and away from your pack will dictate what you take with on the route. For an extreme example, Alex Honnold soloed Freerider in under 4 hours, while a more average party would probably take a portaledge.
Be realistic in your estimates, and don't forget to account for time spent at belays getting racked up for the next pitch. One rule of thumb would be to figure between sixty and ninety minutes to lead and follow a pitch for a party climbing a route that is within their abilities (but not trivially easy) with good rope management and organization at belays. Tack on some extra time if climbing with a party of three.
Are you climbing with a longer rope / willing to simul-climb short sections in order to link pitches? Are you comfortable running it out over easier terrain, or do you like to sew pitches up, frequently placing gear? Does the route have complicated route-finding and require building gear anchors at hanging belays, or is there a line of bolts leading to a bolted anchor on a ledge? Is the route a straightforward jug haul/hand crack, or is it delicate slab climbing?
Don't forget to account for descent time! Doubly so if the descent is intricate.
If it starts getting dark or weather starts building, do you have the margin to accelerate your pace and quickly fire the rest of the route, or is the climbing difficult and already at your limit?
How committing is the route?
Is the route easy to retreat off? If it gets towards dusk, can you just pull the plug and rap back down to the ground off of fixed anchors? Or is the only feasible way off the route to top it out and take a walk off descent?
How long until dark?
One of the absolute easiest pieces of gear to take up a multipitch route is a small headlamp. A bad situation can become truly epic, if not fatal, in the dark. Rappelling into the darkness towards an unknown anchor is a nightmare and the type of mistake you generally don't repeat. A headlamp is easily carried by either clipping or girth hitching it to a gear loop. Some truly small headlamps (e.g., the Petzl e-LITE) can even stash neatly in a pants pocket or the zippered pocket found on some chalk bags.
What's the climbing like?
The nature of the route can make a large impact on how much extraneous gear you're willing to take up a route---off-widths and chimneys are annoying with a mess of extra gear clipped off your harness and downright frustrating with a backpack. While this can be somewhat mitigated by intelligent racking or clipping a backpack off a sling to trail behind you, it is an annoyance that is best avoided if at all possible. In contrast, a 5.7 hand crack or ridge scramble might be almost as easy with a pack on.
What's the weather like?
Even though it is currently a sunny day, always keep in mind that the weather could (probably?) change. Red Rocks, Nevada, is infamous for winds that come whipping down the canyons in the evening and the Rocky Mountains commonly generate afternoon thunderstorms on days that start with clear skies. Will you have a clear view of approaching weather? Would you be able to retreat if you see darkening skies? If the weather did turn, would it just mean being a little bit cold and wet, or would it be an emergency without extra layers?
Consider slightly overdressing, e.g., wearing technical fabrics, pants instead of shorts, and perhaps even an unzipped thin jacket. While it might be overkill if the weather stays nice, it can make the difference if there is inclement weather.
Some wind / rain jackets can stuff into a pocket and have a carabiner loop for clipping it to your harness; you can also find a small stuff sack for the same purpose. Alternatively, you can tie a jacket around your waist---make sure to tie a good knot and "twist" the jacket up to avoid having a large flap of fabric in your way.
Food and Water
Try to tank up on water and eat a snack before starting a route in order to minimize on the amount you need to take with. Snacks can easily be stashed in pockets, while a water bottle is usually clipped of a rear gear loop (soft-sided water bottles are quite nice for this). Make sure that you're clipping into something structural on a water bottle---that strip of plastic on a classic Nalgene bottle is just a leash to keep you from dropping the lid.
Consider whether you will be climbing in the shade or sun and how hot and dry of a day it is (remember that these can change over the course of the day!). Consider sharing a water bottle if you won't need that much water on the route.
On the topic of food and water, give some forethought to the other end as well. Try to take care of business before starting up the route (always follow Leave No Trace principles!). On some popular routes, this is such an issue that you can almost route find by sniffing your way to the next belay ledge...
No. If you're climbing with a pack, you can just dump the rope in there. Otherwise, learn how to backpack coil a rope.
Some people won't go anywhere without a knife, and they can potentially be useful for cleaning up tattered messes of webbing at anchors and dealing with stuck ropes...just no Vertical Limit moments! If you do carry a knife, chose something small and light, either stashing it in a pocket or clipping it off a harness. Consider something purpose built towards climbing like the Petzl Spatha, Trango Shark Nut Tool, or Buck Redpoint, and consider doing something to ensure the knife doesn't accidentally come open (a small wrap of climbing tape can work well). If you don't carry a knife, be aware that it is possible (if tedious) to cut a rope without a knife.
Consider bringing some extra anchor material (~7mm cord or 9/16"--1" webbing) and a few rap rings / mallions / leaver biners for improving rap stations. Try to avoid adding to "rat nests" by removing older tat if you feel its necessary to leave some of your fresher anchor material.
Bring along a few extra pieces of climbing equipment in case effecting a self-rescue becomes necessary. An extra cordalette, a prusik/tibloc, and a couple extra carabiners can be miraculous with a little training and ingenuity.
If possible, try to avoid carrying an entire guidebook up the route with you. Some options include taking a picture of the route description with a digital camera, photocopying the relevant pages, studying the route beforehand and committing the description and topo to memory, and even ripping the relevant pages out. This will also depend on the nature of the route, e.g., line of bolts vs trad route with intricate route finding.
So how do I carry all this crap?
Well, it depends on how much crap you're taking with you! From least amount of extra gear to most, the options would be
- If you only have a couple extra items, say approach shoes and a headlamp, just have everyone clip them to their harnesses.
- Have the follower climb with a small pack. A small lightweight pack or stripping down a climbing pack (remove top lid / framesheet / waistbelt) will work; clipping the waistbelt around the rear of the backpack can do wonders for making climbing and accessing harness gear loops easier. Consider keeping a few items clipped to the leader's harness to share the load. This works if you need, say, a couple liters of water, a few bars, headlamps, and a thin jacket each.
- Have a small leader's pack and larger follower's pack, perhaps by having a water bottle, headlamp, and jacket in each pack, with extra layers and shoes in the followers pack.
- Resort to hauling a pack or haulbag. A method of last resort as this ups the complexity and time taken, but can be necessary if the climbing is truly difficult. One special circumstance would be a route that is generally reasonable to climb with a pack, but features a crux off-width/chimney pitch. Rather than struggling to climb this one pitch with packs, consider hauling a pack through this section. It will definitely be more enjoyable and might even be faster.
- If doing a (long) walk-off descent without backpacks, consider loosening the leg loops on your harness (if possible) in order to make walking easier. Furthermore, try to shorten everything on your harness as much as possible--for example, re-rack cams off the thumb loop instead of the sewn sling. The cumulative effect will lead to less "gear bounce" as you walk and help reduce equipment getting hung up on bushes.
- When done with a climb, consider whether you brought enough extra gear and plan your next climb accordingly. Note that often times, the correct answer is "learn to organize gear and move more efficiently" rather than "bring more stuff next time."