The Father, Son, and Spirit each fully posses the Divine Nature. So whatever is true about the Divine Nature (the Godhead) is true about each person. Each Person is love, is mercy, is truth, etc. Each person is all-knowing all-powerful, etc. What distinguishes them is their relationship to one another.
The Father does not proceed. The Son proceeds only from the Father. The Spirit proceeds from the Father and the Son.
The analogy often used (and just as often misused) is that the Father is analogous to the will, the Son is analogous to knowledge, and the Spirit is analogous to love. In the human person, we have will, intellect, and (hopefully) love. But this is just an analogy. Each Person is Love. Each Person exercises will and is all-knowing.
The above explanations where God loves himself or knows himself are not correct. The Father’s knowledge of Himself is said, as an analogy, to be so full that it is the Son, and the love between the Father and the Son is said to be so full that it is the Spirit. But in fact each Person has will, knowledge, and love.