I think this depends on the child to some extent. My older son never went through that stage at all; he went straight from using the verb alone ("go park") to using I ("I want to go to the park"). He (and his friends at daycare) did go through the you/I confusion as is common; it's possible 'my' is simply a variant of that. It's also possible "my" is really intended to be "I" - they are the same sound root, just with 'm' in front.
In terms of not conjugating the verb properly, that I did see in my child for a little while. "Is" is a common issue with children, particularly if their parents primarily talk to them in the third person ("David is doing well!" rather than "You are doing well!" / "Mommy is going to the store"). Avoiding talking in the third person can help (certainly did for us); it's not going to fix things immediately, but the child isn't going to learn conjugations from a book - he's going to learn them from listening to Mommy and Daddy use the vocabulary.
For my son, he learned to use I/you properly and am/are shortly (a few months perhaps, though they seem closer to me) after he began using "complete sentences". Ie, he went from "you go school" to "I am going to school" more or less in one step; both the you/I and conjugation happened together along with adding other helper words like participles and adverbs.