A few notes from a few lectures on my seminary school on Matthew. I'm Dutch, so please excuse any spelling errors:

Matthew is the only that speaks about "the kingsdom of heaven". The other gospels all speak about "the kingdom of God". This gospel is written especially for the Jews, you can see that for example Jewish rituals aren't explained.

The learnings are in thematical order. The central theme is "The kingdom of heaven":
1-2: birth
3-4 John the Baptist and Jesus
5-7 Kingdom Part 1: Speech on the mountain - Beginning of the Kingdom
8-9 Jesus miraclepower
10 Kingdom Part 2: Sending Out Speech - education and announcing the Kingdom
11-12 Authority and Softness
13: Kingdom Part 3: Speeches of likenesses - hidden growth of the Kingdom
14-17: Resistance and recognition
18: Kingdom Part 4: Community Speech - to learn to coop with eachother in the Kingdom
19-23: Tributing and casting away
24-25: Kingdom Part 5: Speech of the Future - the future of the Kingdom
26-28: Suffering, dying and ressurection

Matthew also quotes prophesies, not as much as to claim he's right, but to place Jesus in the context of the Jews, that it was for them. Baptism - he becomes a reprensentive of the people, desert 40 days - like the 40 years Israel spent in the desert, etc.

He also shows that the heathen people often believe before the Jews (3 wisemen, woman at the well, headman).

Gospel (Greek: euangelion) actually is a much used word, when an emperor had a victory, this was proclaimed as the "gospel". Another word like that is Saviour (Soter), which is used for the emperor as well. The usage of these words by the apostles was a political statement. as well as it was understandable for the people.

Matthew 1:1