I would like to know, if Jesus taught us to address the Father in our prayers (i.e. the Lord's prayer, see Holy Gospel of St. Matthew 6:9-13) why do we also address the Son and the Holy Spirit in our prayers (e.g. in the Agpyea)? What is the Biblical basis for this?
The Lord Jesus Christ Himself taught us that we should pray and whatever we ask in His name He will do and the Father will give. "And whatever you ask in My name, that I will do, that the Father may be glorified in the Son. "If you ask anything in My name, I will do it" (John 14:13-14). "You did not choose Me, but I chose you and appointed you that you should go and bear fruit, and that your fruit should remain, that whatever you ask the Father in My name He may give you" (John 15:16). "Most assuredly, I say to you, whatever you ask the Father in My name He will give you. Until now you have asked nothing in My name. Ask, and you will receive, that your joy may be full" (John 16:23-24).
The Holy Spirit is God's gift to us to dwell in us and lead us in our life. "And I will pray the Father, and He will give you another Helper, that He may abide with you forever; the Spirit of truth, whom the world cannot receive, because it neither sees Him nor knows Him; but you know Him, for He dwells with you and will be in you" (John 14:16-17). To know Him means to have a relationship with Him, to talk to Him and be close to Him. This can only be accomplished through prayers for praying is the only way of communication between us and the Holy Trinity. Once the Father gave us the Holy Spirit in the Mystery of Chrismation, we pray to the Holy Spirit to be with us always and bless us with the fruit of the Spirit.