Briefly define and give scriptures (not that I'm trying to make anybody a dictionary theologian)? Thanks.
I don't have time to do all of your homework for you, but here is a brief sample:
The gospel of the the kingdom is the good news that God was about fulfill His promise to Israel to establish the Messianic kingdom; this kingdom at hand was good news.
Mark 1:15 (KJV)
(1:15) And saying, The time is fulfilled, and the kingdom of God is at hand: repent ye, and believe the gospel.
The gospel of the grace of God is the good news that God is giving the free gift of salvation to all without distinction apart from Israel, the covenants, etc. etc.
Acts 20:24 (KJV)
(20:24) But none of these things move me, neither count I my life dear unto myself, so that I might finish my course with joy, and the ministry, which I have received of the Lord Jesus, to testify the gospel of the grace of God.
You can conflate all of these gospels if you like. Most of churchanity does.
The LORD Jesus Christ was NOT preaching the gospel of the grace of God during His earthly ministry to Israel.
Matt 6:14-15 (KJV)
(6:14) For if ye forgive men their trespasses, your heavenly Father will also forgive you: (6:15) But if ye forgive not men their trespasses, neither will your Father forgive your trespasses.
That is NOT the gospel of the grace of God.
Part of the gospel of the kingdom is HEALING for EVERYONE....
Matt 4:23 (KJV)
(4:23) And Jesus went about all Galilee, teaching in their synagogues, and preaching the gospel of the kingdom, and healing all manner of sickness and all manner of disease among the people.
(NOTE: 'the people' refers to Israel)
Peter and the gang continued this right up until Israel finally rejected the kingdom.
Acts 5:16 (KJV)
(5:16) There came also a multitude [out] of the cities round about unto Jerusalem, bringing sick folks, and them which were vexed with unclean spirits: and they were healed every one.