- He begins to know his surroundings.
- I have begun understanding this lesson since first session.
He begins to know his surroundings.
is grammatically correct, but native speakers would not use this construction. In the present tense, He is beginning to know his surroundings would be normal phrasing.
That construction would normally only be found as a clause in a larger context, in a conditional or some other sentence with additional components, for example:
When he begins to know his surroundings, then ...
As he begins to know his surroundings, ...
The second sentence
I have begun understanding this lesson since first session.
is not grammatically correct - first session requires the definite article (the).
I have begun understanding this lesson since the first session.
However, this construction is also not normal phrasing. A more typical phrasing would be
I have understood this lesson since the first session.
but given that the question seems to be around the use of begin, to emphasize that the commencement of understanding of the lesson is linked to the first session, use the imperfect form began:
I began understanding this lesson in the first session.
"Begin to do" is used for a specific situation.
"Begin doing" is usually used to make a general statement, in which the action that is done is repeated or regular or habitual.
So the first sentence would sound a little bit weird to me, because it miss the part that specify its meaning. For example "As he begins to wake up, he can hear people talking all around him."
And for the second one, it seems correct both grammatically and situationally
"I have begun understanding this lesson since first session". (= I began understanding the lesson as an activity or skill, that is, as something i did regularly and repeatedly since first session.)