You can find a YouTube playlist here with many of the songs suggested below.
Gathering
There are some great popular songs which can link to the general themes of the readings and be used for gathering. Alleluia, Alleluia! Give thanks to the risen Lord (MP 30) would be more often sung at Easter but v3 and 4 in particular are relevant. I know that my Redeemer lives (CH 423) is set to a wonderful Sacred Harp tune in CH4 but there are alternative versions of the lyrics set to the better known 'Duke Street' (Hymnary). Great is thy faithfulness (CH 153 / MP 200) has that more explicit focus on God the Father, but v3 can lead us towards the New Testament readings while Here in this place / Gather us in (CH 623) can preempt themes of new life/light. Come now is the time to worship (CH 196 / MP 1040 / CCLI) is a well known worship song option while Come and worship Christ the King (Resound) is something less well known but is a great upbeat song with a catchy wee vocal hook that makes it memorable. On a similar theme
Word
The psalm doesn’t have many settings and singing it responsorially is probably the most approachable option, either using the popular chant O Lord, hear my prayer (Taizé / MP 908 / PFAS 102A) or the wonderful gospel style Be with me Lord (PFAS 86B / OCP). You are good and forgiving (Resound) could be an option for bands and is described as a song to be sung in the good times as a meditation on God's character and in the bad times as a cry for help.
Response
His eye is on the sparrow (Hymnary) would be a great option for responding to Matthew, a gospel song associated with Ethel Waters and Mahalia Jackson. It can be sung in a simplified hymn version or else in a more contemporary take by Aubrey Assad (CCLI). You could also consider songs about following Jesus such as Will you come and follow me (CH 533) and Jesus you have called us (Resound).
We know that Christ is raised and dies no more (CH 635) or Baptized in water (CH 636) would link well to Romans, with the alternative tune of Bunessan helpful for introducing the latter. Saranam (CH 573) is a beautiful song from Sri Lanka with an Indian melody which would be best suited to solo verses with the congregation joining in the chorus. Were I the perfect child of God (WGRG) is written for a baptism service but has an alternative text which would make it suitable for this service. Holy Spirit (Getty / MP 1183 / CCLI) starts with our personal renewal but moves in v3 into a corporate call.
The God of Sarah praise (GIA) is an excellent rewriting of “The God of Abraham praise”, set to the same tune, which allows the women in the Old Testament story to be brought to the forefront. The Genesis text also offers the possibility of exploring interfaith relations, and Hope of Abraham and Sarah (MV 148 / GIA) is a song about unity among the Abrahamic religions. A common line in Genesis and Matthew is "Don't be afraid" and there are a couple of songs which reflect on this, Do not be afraid (CH 191) and Don't be afraid (Wild Goose / YouTube).
Sending
Some general songs which pick up on themes from today are Lord for the years (CH 159 / MP 428), which prays for our land, our world and ourselves and We know that Christ is raised and dies no more (CH 635) with its wonderful imagery of "the spirit's fission". Living Hope (CCLI) almost takes us on a journey through the readings, starting with the desperation of the psalm, then moving to the theme of sin, before concluding with the new life that comes after the cross. You could also use Union with Christ (New Scottish) or Happy Day (MP 1224 / CCLI) to link to Romans, or Take up your cross, the Saviour said (CH 402 / MP 935) if you have focused on the Gospel.