What Tools Should I Have to Work on a "3 Speed Bike"?

What tools should I have to work on a three-speed bike "from top-to-bottom".

This is a question that comes up a lot, and which has gotten at least a little discussion lately on a couple of bicycle fora online. I don't think it is possible to produce a truly "perfect" list of tools, but it is possible to help someone get started.

This is a long list for the person looking to do 100% their own work. If I had to equip a little shop to fix these bikes, my list would look like this.  This list does not contain the obvious items, like tire levers, bike pump, etc that you should already have.

Rather, this list is for the person interested particularly in older-type utility bikes.
Modern  bikes rely basically on the head-head multi-tool. You see these in bike shops: a Swiss Army Knife-style tool with head heads to tighten any screw or bolt on a bike. With these older bikes, you can't get by with just that tool.

The items underlined are "must" haves, in my opinion. The items without underline are nice to have, but not necessary.

Wrenches and Pliers

• A set of good adjustable crescent wrenches: look for forged rather than cast wrenches, especially older US- or British-made. Make sure the wrench jaws are crisp and not chewed up. Get several sizes.  I have one very old, "Utica" brand crescent that belonged to my great-grandfather that I especially love. It's exceedingly well-made.
• One Channel-Lok, wide-mouth adjustable crescent : Channel-Lok makes a wrench with VERY wide jaws that comes with a blue rubber handle. This wrench is wide enough to handle the very wide headset nut on a bike. If you don't have this, you're stuck with big pliers.
•  Two small sets of combination wrenches in imperial and metric sizes: crescent at one end and box wrench at the other. Again, look for US- or British-made forged wrenches. I'm partial to old-school US Sears Craftsman.
• What about "Whitworth Wrenches"?: Whitworth is an oddball sizing that's not metric or Imperial. The adjustable wrenches usually will be fine for this stuff, but if you do see Whitworth wrenches near you for a good price, pick them up. They help but are not totally necessary. 
• A Sturmey Archer cone wrench and a good quality pedal wrench. These are the very flat, specialty wrenches used for those specific purposes. Both are life-savers when you need to adjust a cone or tighten pedals.

• A set of good, forged pliers (e.g., Channel Lok, US-made pliers). These can be helpful to grab things, but if you're just doing a couple bikes you may not need them. 
• A good set of cable cutters or bike cable nippers: you need something sharp, with a good edge to cut brake cables.

 Socket/Nut Drivers

• Consider investing in a good socket set. I like Craftsman, but Napa and similar are good. Get the "nut drivers" with them -  those screw driver handle attachments that tighten down nuts and bolts using the screw driver handle. The nut driver is great for tightening brake blocks.

 Screw Drivers

• You'll want a good, complete set of well-made screw drivers in flat blade and Phillips heads. Make sure the blades are not all chewed up. Get drivers with nice, firm rubber handles. 

Hammer/Mallet/Punch

• Maybe a good rubber mallet and certainly a good ball peen hammer. I prefer the ones with nice, hickory handles. The ball peen allows you to hammer out fender dents, and the rubber mallet is a good "knocker" for stubborn stuff (where appropriate!).  The ball peen should be ball on one side, flat striker on the other.
• A set of punches: get good, hardened punch like those that Bosch sells. These are used to loosen the ball ring on the Sturmey hub and to loosen up bottom bracket cups.

Other Tools/Supplies

• The Bikesmith Cotter press: get this cotter press if you plan on working on bottom brackets at all. You can use a C-clamp and socket piece but the Bikesmith press is the best tool for the job. It works really well.
• Calipers: not totally necessary but great for measuring unknown screws etc. I have a simple set of vernier-style calipers for a few dollars.
• Grease/grease gun: invest in a decent lithium grease at the very least. You'll need grease. A small grease gun is nice (not totally necessary) for getting into cups.
• Dremel rotary tool: if you plan to do more than one bike, or work on multiple bikes a Dremel can be great for clean-up/grinding/etc. Don't go cheap, get a real Dremel. 
• 20 Weight Oil, Like 3-in-1 motor oil. You'll get used to looking for the little, blue bottle. Three speed transmissions on old bikes live on this stuff.
• Penetrating oil, like Kano Kroil: this helps when you have something rusted out and frozen in place. Trust me, it WILL happen to you sooner or later. Grab some WD-40 as well while you're at it.
• Bronze or copper wool and/or brass bristle brushes: great for cleaning rust off of plated parts.
• A small torch, such as a butane torch to use with the Kroil to free frozen parts is nice to have, but not a "must" unless you're doing a bunch of projects or stuff that's been sitting outside a long time.
• A tray of sundry nuts, bolts, and screws: you know, the trays you see in the hardware store with a bunch of differently sized nuts and bolts. These can be used to replace broken or stripped parts.
• Canned air: normally thought of as a computer tool, this stuff is great to blow out old dust
• Polish: great for shining up paint. Try Simichrome for chrome polish and any good quality car polish for paint. Remember to go easy with it though.
• A small bench vise: even just a clamp-on hobby vise will do. You just need something to hold parts steady while you work on them.
• A set of hex keys/wrenches: sometimes people throw head-headed screws onto old bikes when they rehab them. I have one bike where someone replaced all the chain guard screws with modern, bike-type head head screws. If you come across one, you'll be able to get it off of the bike.
• A thin tipped but strong magnet: great for pulling out bearing balls from races without dropping them. Reach in with the magnet and pull them right out. If the magnet is strong, you shouldn't have to worry about dropping them.