Both the words "forgiveness" and "remission" are translated from the same Greek word, aphesis. The word literally means a release, a grant of freedom. It is used extensively in the Greek old testament in reference to the "year of release", the "release" of debtors from their debts, etc. It is also used in reference to the "release of waters" in several places. It is also found in Isaiah's prophecy that Christ quoted in the synagogue:
The Spirit of the Lord GOD is upon me; because the LORD hath anointed me to preach good tidings unto the meek; he hath sent me to bind up the brokenhearted, to proclaim liberty to the captives, and the opening of the prison to them that are bound;
(Isa 61:1)
The word "liberty" in the Greek version is aphesis, release.
The Spirit of the Lord is upon me, because he hath anointed me to preach the gospel to the poor; he hath sent me to heal the brokenhearted, to preach deliverance to the captives, and recovering of sight to the blind, to set at liberty them that are bruised,
(Luk 4:18)
Here, the word "deliverance" and the word "liberty" are both aphesis, release.
A release of sins means a release from the penalty and curse and power of sin. This is signified by both the word "Forgiveness" and "Remission". Remission is technically a little closer to release, but in regards to sin as incurring a debt that results in bondage forgiveness is equally appropriate.
There is no verse I know of that uses both forgiveness and remission as if they were two different things. The idea that forgiveness occurs at repentance and remission occurs at baptism arose out of the UPC formation which was a compromise between two groups of Oneness Pentecostals. One group held that people were saved (forgiven of their sins) when they made their first trip to the altar in "repentance", while the other group held that this did not occur until they were baptised in the name of Jesus. A compromise was reached for organizational unity, which developed into a doctrine of salvation where a person first get forgiven at the altar when they "turn to Christ" or "Accept Jesus as their Lord and Saviour", then they get remitted when they get baptised in His name, then they make a second trip to the altar for the Holy Ghost.
Biblically though, the general and normative pattern is repentance, baptism, and receiving the Holy Ghost happens pretty much all at once when a person hears the Gospel, and decides to become a disciple of Christ signified by baptism, and receives the Spirit more or less immediately after coming up out of the water. And yes, there are some exceptions in the Bible to the pattern, but the exception proves the rule, as they say.