Excellent question Otis-san.
First, *'Watashi'* is used for both _"I"_ and _"me"_ by adult speakers of both sexes.
*'Watashi __no__'* translates as _"my/mine"_.
*'Watashi __mo__'* translates as _"me too"_.
Now lets take close look at the two connecting particles, *no* and *mo*.
:idea: __*の* *(no)* _"apostrophe 's"___
When the particle *の* *(no)* is placed between two nouns, it makes the first noun into a modifier of the second. When the first noun is a personal name or a noun referring to a person, such as *Watashi*, it generally describes the following noun (item) as belonging to that person - indicating POSSESSION.
For example, *Watashi __no__ ie desu.* = Literally translates as _"I - 's - house - is."_ Meaning _"It is my house."_
The noun + *no* combination shows POSSESSION.
:idea: __*も* *(mo)* _"also"___
*も* *(mo)* marks something as being in addition to something else.
Thus, *も* *(mo)* indicates that the noun it follows is connected in some way in addition to something or someone else that has been previously mentioned in the given context.
*_[A specified item or person] + も = [A specified item or person] too / also / as well_*
Suppose your colleagues are planning on attending a business conference, you think you will join them, and politely say...
私も行きます。 *Watashi mo ikimasu.* _"I will go too."_
So think of *も* *(mo)* as the INCLUDER, and *の* *(no)* as POSSESSOR.
I hope this has cleared up some questions!
Watashi no vs. Watashi mo