Changing a quartz watch battery is easy, you just have to know how
to open the case (see my How To on that), have the right battery,
and have the right tools.
There are three types of tools used by people to change watch
batteries, whatever is at hand, kinda ESD safe tools, and real ESD
safe tools. First we need to discuss ESD so you know what you are
dealing with. ESD stands for Electro Static Discharge and is
basically what happens when you walk across carpet and touch a metal
doorknob and get the mess shocked out of you, that's over 3,500
volts there buddy! For reference your quartz watch runs off either
1.5v or 3v. Needless to say that much voltage can easily fry the
electronics in a quartz watch.
ESD is caused by many things, walking across carpet, your clothing
moving around on your body, dry air inside a climate controlled home
or office, etc. The problem is most people equate static shocks with
carpet and doorknobs (metal) so they assume plastic is safe. Try
pulling Styrofoam cups off the stack inside the plastic bag they
came in, place your arm near the stack as you pull the cups off the
stack, the hair raising on your arms shows you the static
electricity buildup. You may also not know that cling wrap clings by
using a static charge. Now we know that just because something is
plastic, or non-metallic, that does not mean it will protect the
delicate circuits inside your watch (generic plastic tweezers you
picked up at Wal-Mart are not a great idea).
So what do you use to work on quartz watches? Tools from reputable
manufacturers like those above that are designed to be ESD safe. But those are
expensive right? That depends on what you term expensive. I use two
sets of tweezers for quartz work, one $40 set of high resistance
carbon fiber replaceable tipped ones, and one $6 high resistance
solid carbon fiber. I use two for a variety of reasons but the
latter will work just fine. If for whatever reason you need to use a
screwdriver in the case, both Wiha and Excelite make real ESD safe
sets which are fairly inexpensive. These screwdrivers are made from
two different types of high resistance plastics and then the metal
blade, this keeps any static away from the part you are working on.
Also note in the picture above I have a blue plastic probe, designed
for use in electrical circuits, in case I need to push or pry on
something.
Lastly, I use a special ESD safe work mat and discharge myself
before touching anything. You can usually discharge yourself by
touching anything grounded such as a metal water pipe, metal gas
line, bare screw head on the center of an electrical outlet, etc. I
have a grounding plate plugged into a grounded electrical outlet
just for this.
Now assuming you are discharged and have the proper tools, replacing
that battery is a snap! Most batteries these days are not even held
in place once you get the back off, just carefully lift it out and
place the new one in. Some have a sliding tab on top of the battery
that swings out of the way so you can lift out the battery. The last
type is the ones where you need to remove a screw (with the correct
ESD safe screwdriver of course) on an arm over the top of the
battery. There is nothing more to it.

The watch above has a little metal arm that comes over to secure the
battery. To move this arm you need to remove the screw below and to
the left, just to the right of the copper coil. Once that is done
and the retainer is removed, you can just use your tweezers to
remove the battery like this:

Just reverse the procedure to put in the new battery and you are
done!
Before you throw up your hands and take it to Wal-Mart to have
someone there do it because they know what they are doing, I read a
story by a well known watchmaking instructor about an employee at
Wal-Mart prying the back off a $20,000+ watch (which was not a snap
on back), destroying the back, the springs that held the back on,
the movement, and everything, only to find out the watch was a
mechanical and did not have a battery at all! The people running
these places have no idea what ESD is, what an expensive watch is,
or how to do anything. They do not train their employees on how to
do anything, much less the correct way to do it. It is much better
that you do it since you will at least have some respect for the
watch.
Good luck!