OBVIOUSLY you can't just start turning screws...there are a number of things you will need to do (and get) in preparation. You will have much better results, and a much more enjoyable experience if you take the time, effort, and expense to get most of the items listed below BEFORE starting. Keep in mind that these are just the bare essentials... a serious repair person has hundreds of other specialized tools... but we'll save that for another discussion. The bottom line is that preparation and having the right equipment is vital to success of any repair job!.
Workspace
Set up a well lit workspace with a clean, flat surface. For a beginner a kitchen table will do just fine, but as you become more serious you may want to consider a watchbench made specially for repairing watches and a good fluorescent lamp which is easier on the eyes. A watchbench has small built in drawers for all of the tools and parts you are sure to accumulate over time. The design is tall, with a surface that (when sitting) is close to the face so the repair person doesn't have to slump over when working. Some repairmen will put cork or linoleum on the surface. These soft materials stop parts from bouncing away, and aren't hard enough to damage anything dropped on them. Some benches even have a slide out "parts catcher" that presses against the belly when sitting. Mine is a wood frame with a loose piece of canvas tacked to the bottom. When a part is dropped and goes "over the edge", this brilliant contraption catches it!
Most beginners will (as mentioned above) begin at a kitchen table. A good tip for working at a table is to lay down a fine piece of white fabric (like a plain white pillowcase or cheesecloth) to do your work on. The main advantage for doing this is if you drop one of the parts, it won't bounce away and become lost. I suggest white because tiny screws (etc.) are easy to locate. This practice will also save your table from scratches, and protect your watch from the tables hard surface. To keep it from shifting, you might want to consider taping it down with masking tape as well.
Screw-driver set
You WILL need a good set of watchmakers screwdrivers. This is a MUST. The sets found at the department store for working on eyeglasses etc. will cause more trouble than they are worth. They are not precision made and you will easily damage or strip the highly polished screws found in most watches. The keen eye of a serious watch collector will pick this up immediately; it detracts from the look of the watch thus reducing value of your watch. Track down and buy a set of watchmakers screwdrivers. Even a beginners set will be much better than anything you can find at the dollar-store.
The proper way to hold it (Right Below) is to grip the screwdriver shaft between the thumb and middle finger, with the index finger on the cupped end (which rotates). Press down on the top of the screwdriver and swivel it between your thumb and middle finger; turning the screw.
Parts Tray
You will also need a place to put your parts as you remove them from the watch. Don't get in the bad habit of placing parts on the table or bench (trust me on this one!) you are bound to lose something. I would highly recommend a common (inexpensive) flip top tackle box. Get one with 8 to 12 sections. They are available in most department stores. An empty egg carton will also do just fine, but will flip easily sending parts flying, so be careful. Simply place the parts in order from left to right as you remove them from the watch. When it is time to reassemble, just go in reverse order and you will know what part is supposed to go next. Numbering the compartments is a good idea to make absolutely sure you are going in proper order. The parts trays shown here are great if you have more than one job going at once- they are compact and stack-able...
Tweezers
Get a good pair of fine tweezers. Removing and returning small parts without tweezers is nearly impossible without them. It will take a while to master this skill... use a soft grip... hold something too tight and the part will ricochet around the room... Tweezers are available at the department store, but not nearly as accurate as a good set you will eventually want fine quality watchmakers tweezers.
Cleaning Supplies
If you get really serious about this hobby, you will definitely want to buy a mechanical or ultrasonic parts cleaner and specially designed parts solution. You can buy the old bulky mechanical ones on Ebay for a few hundred $$ and the solution costs about $50 per Gallon. The ultrasonic machines are more expensive. Both machines use the same liquids. This is definitely the way to go if you get really serious about the hobby.
When I first began this hobby at the kitchen table I used Naphtha. Naphtha can be found in most hardware stores. I have found that it works very well and evaporates leaving almost no residue. Simply pour a tiny, tiny bit into a small cup and soak your parts for a few minutes. Then brush between gears and in jewel holes with a small coarse tipped paintbrush, or eyelash brush. Once you are satisfied the part is clean, remove it from the solution, place it on filter paper which absorbs most of the fluid. The remaining fluid simply evaporates leaving a clean part. Use a blower to quicken the dry time.