The type of battery required is printed on the back of the case. In this case, I was looking for a 377 type watch battery for both watches. They came in a two pack from the drug store for about $7 or $8.
Once the case back is off you can usually see the watch battery in the back of the mechanism. If you look carefully with a magnifier, you may notice that crosses onto the back of the battery. When you pry out the battery, be careful not to bend that out of shape. It's a little flexible to enable you to get at the battery out, but it's necessary to get electricity out of the battery.
Next slip the new battery in where the old battery was. Place the watch caseback on the back of the watch. Be careful that any indent on the caseback lines up with the winding (setting) stem that comes from crown on the case so that you don't bend that stem. Then you press the caseback down onto the case. Sometimes that can be difficult to do by hand. That's why it is useful to have a watch caseback press like the one pictured below.
There are discs for the press that match different watch sizes. You'll want to pick the size that matches your watch and place that on the upper piston that gets pressed down on your watch.
Then, you put your watch and carefully lined up caseback on the press and gently press down. It should snap right into place. Be careful about putting too much weight on it or you could damage the watch crystal. It didn't take much force at all using the press. If it doesn't work, don't force it. Check to see that the caseback is lined up and that the disc matches your watch size appropriately.
I was able to get both watches back running in just a few minutes. Please note that I am just an amateur watch collector on the internet. If you have very valuable or heirloom quality watches, you should probably take your watch to a jewelry store for a watch battery replacement.