Oil and Oiler
Specifically designed oils for watches are another must. I would recommend Moebius. You don't need to buy a ton of it either. One drop is enough to oil dozens of watches. There are a number of different oilers available. A basic oiler is a thin piece of wire with the end flattened slightly. It is dipped into oil each time a drop of oil is needed. They come in different thickness' to deliver different amounts of oil depending on the job. Also shown below is an automatic oiler. These are a thin glass tube full of oil that delivers one tiny drop each time it touches a surface (like a jewel hole). These are convenient because dipping is not necessary.
Loupe
You need to see exactly what you are doing and therefore you will need magnification. A jewelers loupe is a must. There are different variations to a standard loupe that you should consider. Over the years I have become partial to eyeglass clip on magnifyers (shown below) that I use with a set of reading (cheaters) glasses. Some watchmakers also use some 'Rodico' to attach a loupe directly to their glasses (also shown)..
Blower
Physically blowing on a watch movement is not the best idea. As unsavory as it may seem, when you blow on something with your mouth, you actually blow particles of spit onto the object. I would recommend buying a blower. Not only are they cheap, but they are great for drying off parts that have just been removed from solution, or for removing dust from a dial.
Hands
Remover and hands pusher. Hands remover tool it a tool designed to lift the hands vertically off of the posts where they firmly rest. The tool avoids damage to the watch dial and the hands themselves. When re-installing the hands, it will be necessary to push down on them firmly with something. A hands pusher tool has a hollow center to avoid damage to the post. Excellent tools that you will need to get the job done right.
Watch paper
This is special tissue paper is used for placing parts to dry. This paper is lint-less and very absorbent. (Scotts toilet paper also works very well, and will do for now)
There are many many other tools that you will want to get as you get more serious about this hobby, but the tools listed above are a good start.
Now that you have the prerequisites out of the way, you are ready to go. I would strongly recommend that you start with a watch that you are willing to ruin, because chances are you will most likely destroy your first watch. Find an old inexpensive 5 to17 jewel watch at a flea market or on eBay. See my section on eBay on how to find a good candidate. Because of their size, pocket watches are easier to work on. I would NOT spend much more than $10-20 on this one!. Do your best to find one that runs but doesn't keep time or stops. This probably means it just needs a cleaning which is what we are learning how to do. The oil in watches breaks down over time and becomes gummy. This slows the watch down, and eventually it stops. The task at hand it to take the watch apart, clean it, oil it, and reassemble it. Easy right?
"YOU WILL MOST LIKELY BREAK THE FIRST WATCH YOU TAKE APART
DO NOT PRACTICE ON YOUR FAMILY HEIRLOOM"
Opening the Back of the Watch
Pocket watch and wrist watch cases have been designed in MANY MANY different ways. I will try to describe all of the ways to open watches, but only from experience will you truly learn how to "crack every nut". See my section on Opening a Watch
Some pocket watches have backs that are on a hinge. You simply place a fingernail under an edge and pop it open. Sometimes you will find another "door" inside. This is a dust cover. Owners would use the first door as a way to stand the watch on a table or night-stand and the second door kept gunk out. The same method is used to open this door... pry a fingernail under the edge and it will usually pop open. A case knife (shown below) can be very helpful in opening may different watches. They aren't sharp enough to cut you, but are designed with a nice rounded edge that makes opening a watchcase a breeze, and won't damage case of the watch. I would highly recommend this purchase... I once slit my thumb open nearly to the bone using a sharp pocket-knife... what a lesson that was!.
Many early wrist watches also had snap on covers. They usually had a lip that indicates where to pry. Carefully place the case knife under the lip and gently pry the back cover off.
If you have a pocket watch that has no lip, or hinge, but appears to have a cover, then your watch may have what is called a "screw back" case. In this type of case, the back is threaded in place. Carefully try to unscrew the back (use a motion similar to opening a jar of pickles).
Wrist watches also commonly have screw back cases. Unlike pocket watches that can be opened by hand, wrist watches have six notches around the edge of the case back. These are meant for the tool shown below. You adjust the wrench so that the three teeth fit into three of the grooves of the watch-back (shown below). Then just turn the wrench and open the back.
There are also pocket watches and wrist watches that can only be accessed through the front. These are the hardest to navigate, and to describe (but I'll try). In a pocket watch one type of this is called a "swing out" case. The entire movement is on a hinge that closes into the back of the watch, and the crystal and bezel hold it all in. If you have one of these... unscrew (or unsnap) the front of the watch (glass side) revealing the dial and hands. You may notice that inside there is a tiny hinge. Pull out the crown "click" (as if you were going to set the watch's time), and see if the whole movement swings out freely.
In some wrist watches you will need a "crystal lift" (shown below) to remove the crystal and then the movement. A crystal lift has many small fingers that grab and constrict the crystal so that it can be removed easily. You may later ask... "how can I take out the movement with the winding stem still attached?" Good question... the stems in these types of watches are two pieces. One piece slides into the other like in a jigsaw puzzle. With practice you will learn to separate them a little jiggle here, and a little jiggle there. Let's just hope your first watch doesn't have something so complicated....