Box Acquisition: Your local bike shop is the best place to start. Boxes are usually discarded and most bike shops will gladly steer you towards the dumpster out back. Bicycle boxes come in several sizes, so get the largest one you can and it is also a good idea to get a second box to cut up and use for padding. If you have 2 pieces of metal rubbing, it will scratch, so put some cardboard in-between. Often, bikes come packaged from the factory with all kinds of great padding to cover the frame tubes, seat and fork & axle ends. You can recycle these, and it makes your packing job much easier. Bubble wrap is also great for padding, but can be very expensive to buy retail. Check out furniture and appliance stores, they are a gold mine for packing materials. I should also mention packing tape, don't buy cheap tape. This is a perfect example of "you get what you pay for". Use the cheap stuff in the box to secure a piece of bubble wrap or padding, but not for sealing the box. Use the good stuff here, and don't be stingy. Those shipping guys do not care about your box, so it's up to you to pack and seal the box securely.
Disassembly: Most bicycles from 1900 thru 1969, (not all, but most) need very basic tools for disassembly. A large flat head screwdriver, a small adjustable wrench, a large adjustable wrench, a hammer, and a thin 5/8" open end wrench (for removing pedals), is all you need. Also, keep a rag handy because you'll need it. I also always keep those disposable latex gloves around, these keep your hands clean, but I always seem to get a little dirty anyway.
First thing to do is remove the pedals. Pedals have different threads and usually they are marked left & right where the axle threads into the crank. My rule of thumb about removal is: which ever side of the bike you are facing, put the wrench in from the top side of the pedal, and the axle always loosens towards the rear of the bike. Got that? Here's another way to explain it: The left pedal loosens clockwise, and the right pedal loosens counterclockwise. Take the pedals and tie them together with string or tape or a tie-wrap and put them in some bubble-wrap, because they can bounce around the box and cause some damage.
Next, remove the seat and handlebars. Remove the seat with the seatpost still attached, then loosen the seat post clamp on the seat and tuck the seat post under the seat and tighten snugly. This makes it easy to pack it in the box with the rest of the bike. Also, wrap the seat with some padding and put it in a shopping bag, then wrap some packing tape around it to protect the cover from rips.
The handlebars should be removed together with the gooseneck. To remove the gooseneck, loosen the bolt that goes straight down the middle of the gooseneck several full turns (4 or 5 full turns), then lightly tap the bolt with a hammer (use a piece of cardboard or wood to protect the top of the bolt). The bolt should drop down and the handlebars should now be loose enough to pull out, you might have to twist the bars from side to side while pulling up. If the bolt does not drop down, try holding the front tire between your knees, and turn the handlebars from side to side, then try tapping the bolt again, (you might have to hit the bolt a bit harder this time) Don't hit the bolt too hard, because you will damage the threads. If, after all this, you cannot seem to get the bolt to drop, just loosen the bolt all the way out, and remove it completely. Then remove the handlebars. The bottom gooseneck bolt will stay inside the fork for now.
Loosen the handlebar bolt on the gooseneck, and turn the gooseneck so it is parallel to the handlebar grips, then retighten. This will make it easier to fit into the box with the rest of the bike. This is a good time to wrap the handlebars with newspaper or padding, because these tend to rattle around the box, and you don't want them damaging any paint or denting any sheetmetal.
Before you remove anything else, this is a good time to pad all vulnerable parts. If the bike has a tank, headlight, taillight, rear rack, these are exposed to possible damage, so wrap them with bubble-wrap.
Also, and this is very important, (my own patented idea!) Loosen the axle bolts for the rear wheel and remove the fender braces from the axle, then put the bolts back on the axle and tighten them. This protects the fender and the braces. If the box is dropped on the end with the fender, the fender will simply flex against the tire and receive minimal damage if any. If the fender braces were still attached to the axle, and the box is dropped the same way, the fender will dent and the braces will bend. If the bike has a rear bookrack, leave the rack braces attached to the axle. Rack braces are usually much stronger. In some cases, I will remove the rack completely and pack along side the rest of the bike. Schwinn racks are week (especially 9-hole racks) and bend quite easily behind the braces, so I usually remove these. Also, loosen the fender braces from the front axel, for all the same reasons.
Take your large adjustable wrench and remove the fork nut, then remove the washer and rest of the hardware and remove the whole front fork. Leave the wheel and fender attached. Then put all the bearings and hardware back on the fork tube and wrap it with a plastic bag or newspaper (usually pretty greasy). Then pad the fender (especially the tips) with bubble-wrap, and put it aside for now.
Now you should have just a few main pieces to pack in the box. The main frame, the front end, seat, handlebars, rack and pedals. Get your box and check the seal on the bottom. Add some tape if you need to (you can never have too much tape).
Ready to pack: Let most of the air out of the tires (not all, leave enough to act as a cushion), this helps save a little bit of space. Then put the main frame in the box first, rear wheel first, then slide the headtube against the other side of the box. This should fit snugly, and should slightly bend the end of the box where the headtube rests. Add a small piece of cardboard here to reinforce the side of the box. Now fit in the front end with the tire at the opposite end of the box as the rear tire. If your front fender has a headlight attached, put that side of the fender facing up. Now slide a piece of cardboard in-between the frame and the front end to cushion from scratching. Try closing the box. A tight fit is fine, you may have to push down to compress the tires a little bit, but this is perfectly ok. If you can't close the box at this point, either rearrange the contents, or find a bigger box. If you can close the box, then fit the rest of the items into the box, they are relatively small and should find spots. If the items have plenty of room, and are "swimming", add some crumpled newspaper as stuffing. Seal up the box. You are done! GOOD LUCK!
PS. Remember to insure your package!