You can find a YouTube playlist here with many of the songs suggested below.
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 and instrumentation and also picks up on words from Isaiah, as does Glory be to God the Father (CH 110). 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 while This is the day (CH 194 / MP 691) has various versions but would work for this day if you use the version with the verses “This is the day… that the Lord has made / the he rose again / when the Spirit came”.
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.
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 / Wild Goose), the latter based on a text by Julian of Norwich, something worth remembering when people say feminine imagery for God is a modern invention.
As it did last week, the Gospel focuses on the “Spirit of truth” and we can link to this in hymns such as Spirit of truth and grace (CH 608), Come, Holy Ghost, our hearts/souls inspire (CH 598) and Great and deep the Spirit’s purpose (CH 617) or the wonderful Tanzanian song Gracious Spirit, hear our pleading (CH 613) which can of course be sung a capella or with simple hand percussion but also works wonderfully with a Gospel feel. Spirit of God (Resound) is a simple three part round which would also tie into this theme.
The Epistle covers a lot of ground and it’s a challenge to find songs which capture all of it. Some songs which could link in different ways include By faith (MP 1262 / CCLI / Getty), We sing the praise of him who died (CH 405 / MP 738) and My hope is built on nothing less (MP 473 / Hymnary) or the two great reworkings of it - the Kenyan song Kwake yesu nasimama (GSW 34 / GIA) and the popular worship song Cornerstone (MP 1334 / CCLI). Holy Spirit, fill our hearts (CH 611) and Come, Holy Spirit gracious heavenly dove (CH 758) would specifically link to v5 and God’s love being poured into our hearts through the Holy Spirit.
The Old Testament reading explores wisdom which is an underrepresented topic in congregational song but The perfect wisdom of our God (MP 1353 / CCLI / Getty) and Who can measure heaven and earth? (Jubilate) are good options here which also touch on the creation theme. You could also consider How happy are all they who hear (CH 686) from the Scottish paraphrases, Come and seek the ways of wisdom (MV 10) and She moves where she wishes (A song to wisdom) (link 1 / link 2 / link 3) which all pick up on the feminine descriptions of Wisdom found in Proverbs.
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).