Deliverance (13a): The second act of the Father for which we thank Him is that He "rescued us." There was a real danger from which He delivered us: "the domain of darkness." Unlike "the light" (1:12) where saints dwell, we were in "the darkness," living in sin! Moreover, we were held in its "domain," under its power. Sin was our master, and hell was our future. But the Father pitied us and snatched us, literally, "out of [ek (ἐκ)]" that destructive place!
Transfer (13b): Now, the goal of this deliverance was not a moral/spiritual autonomy. We weren't rescued from sin to do as we please. Our freedom from sin was joined by ("and") a transfer to "the kingdom of His beloved Son." This means we aren't free agents choosing our own philosophy of life and purpose for life and moral standards and affections, etc. Our Rescuer, instead, placed us in a monarchy, where we are to give ourselves to serve and honor our King!
Redemption (14): There is something striking about this King. He paid for His people's freedom ("the forgiveness of sins"). And the ransom price was His own life! The King loved His people! What's the outcome? His subjects love Him back. We delight in our Ruler! His sacrifice constrains us to love Him affectionately and serve Him willingly. And this was all the Father's plan!
 1:13 begins with a relative pronoun that continues the Father's actions: "who rescued" (cf. NASB footnote).
 This is literally "the darkness" contra "the light" (1:12) where saints reside. This contrast and the mention of sin (1:14) make sin the logical referent. Paul elsewhere uses "darkness" in this way (Rom. 13:12; Eph. 5:11).
 NASB's "domain" is literally "authority [ex-ou-si-a (ἐξουσία)]," which is the "power" (NET, NKJV) to control.
 Sin's authority/power is like that of the master to his slave (Rom. 6:14-17). Cf. also John 8:34 and Titus 3:3.
 "Rescue [rhu-o-mai (ῥύομαι)]" implies danger (cf. 2 Cor. 1:10; 1 Thes. 1:10). In 3:6, it is "the wrath of God."
 "Redemption" is the Greek word a-po-lu-trō-sis (ἀπολύτρωσις) which is "release on payment of ransom."
 "In whom" makes Christ the means of redemption: the ransom, which is His death on the cross (cf. 1:20-22).