You can find a YouTube playlist here with many of the songs suggested below.
Gathering
For setting up the Trinity theme the classic opener would be Holy, holy, holy, Lord God almighty (CH 111 / MP 237) which can be accompanied in a whole range of styles. Today I awake (CH 211) explores our relationship with each member of the Trinity and has some wonderful imagery with a tune that is well worth learning while Source and Sovereign, Rock and Cloud (CH 133) does the incredible task of giving twelve names for each member of the Trinity set to the majestic tune Aberystwyth. Our God saves (CCLI / MP 1192) is explicitly Trinitarian in the way few songs of its style are while In the name of God the Father (Resound) is a less well known song which also does this. Looking at shorter songs Jesus lead us to the Father (Resound) is fast becoming a modern classic which can simply be sung in unison but when used as a round can really build a wonderful time of praise.
Word
There are two excellent metrical settings of the psalm to choose from, either the older language of the Scottish Psalter in How excellent in all the earth (CH 4) or the contemporary language version, O Lord, our Lord, throughout the earth (CH 5), set to a wonderfully upbeat Scottish folk tune. Majestic (CCLI) could be an option for bands while there are a great selection of responsorial options, with How great is your name, O Lord our God (PFAS 8E / GIA) the most concise while O Lord, our Lord, how excellent is thy name (PFAS 8C) and O Lord, how excellent (PFAS 8F) both have a wonderful gospel feel.
With the length of the Genesis text you might also consider splitting it up with a sung response after each day. The same small part of the psalm mentioned above could work, or else a more general option might be an Alleluia. There are lots of great options for this and it could be an opportunity to sing in simple parts (CH 345, CH 751, CH 752).
Response
There are some general Trinitarian songs which might be useful for this week such as O threefold God of tender unity (CH 114), Loving Creator (CH 116), Trinity song (CCLI) and Mothering God, you gave me birth (CH 117), the latter based on a text by Julian of Norwich, something worth remembering when people say feminine imagery for God is a modern invention.
The Gospel focuses on the Great Commission and this is explicitly picked up in Go in grace and make disciples (CH 682) and Mission’s Flame (CCLI). Given the call is to “make disciples of all nations” it could also be good to recognise this as a universal calling to mission rather than seeing ourselves as the centre, and you could use songs of mission from other countries such as Murassala (WGRG / H 82) from South Sudan which calls on us all to be ambassadors of Jesus, or Sent by the Lord am I (CH 250) from Central America.
Songs about peace, reconciliation and unity would link well to the Epistle such as Ososo / Come now, O Prince of peace (CH 275), Put peace into each other’s hands (CH 659), We are one in the Father’s love (CCLI), Lord, make us servants of your peace (CH 527) and Make me a channel of your peace (CH 528 / MP 456) with the latter two both based on the Prayer of St Francis. May the Grace (Resound) is also worth mentioning as it’s one of the few songs which sets v13 explicitly, also adding the Aaronic Blessing.
Uyai mose / Come all you people (CH 757) is a great option to link to Genesis and you could also make it Trinitarian by adding verses where you change “Maker” to “Saviour” and “Spirit”. Is this the way you made the world (CH 242) and She sits like a bird / Enemy of Apathy (CH 593) are some contemporary hymns you could consider, while From life’s beginning / Let praise resound (Resound) and Let there be light (CCLI / YouTube) are good options for bands.
Sending
There are some wonderful Trinitarian hymns set to older tunes which can provide a suitably heightened conclusion such as I bind unto myself today (CH 639) or God, whose almighty word (CH 112). This is a more challenging theme to find in well-known contemporary songs but is picked up in How great is our God (CCLI / MP 1227) which is often followed with a chorus of How great thou art (CH 154 / MP 506). You could also use any of the songs suggested for the Gospel reading if you are focusing more on the call to mission, while Here and now we’re bound together (Jubilate) would also link well to this, as well as the calls to peace and unity in the Epistle..