And they would catch a plane and play on the coast
They do, they call themselves Rush.
And…Satan with a killer harmonica solo.
The Father is the lead singer, front and center, silky stoles tied to the mic, long hair flowing with awesome wind machine effects…the groupies…lets not even start…
The Son is on lead guitar- very much a driving force, weaving in and out melodically, knowing when to take the center stage- very much like Jimmy Page or The Edge
The Holy Spirit-keyboards. Because you cant see it but you know its there
They’re not really a drum band…a little progressive, very new wave though…
Sorry but my band is new wave…
According to Katie Melua it’s God On The Drums, Devil On The Bass
Which makes sense, as how can you have Good without Evil?
And Jesus on Lead Guitar of course… with the Holy Spirit on Rhythm because nobody can hear him or knows what he does, but apparently you’ll hear the difference if he’s not there.
Interestingly, a Google search for holy trinity band returns a surprising (yet IMHO appropriate) result. 
Presumably they would play different instruments for different musical styles.
For example, we have it on good authority that Heaven’s Skiffle Band (link is to a YouTube video) features the lineup:
*“With God on the banjo, Jesus on kazoo
If you don’t believe me well God Bless You!
You got Moses on the jazz horn, just doin’ his thing
Yeah Hallelujah, hear those angels sing!”
*
[Chorus from “Heaven’s Skiffle Band”, written by Captain Skiffle.]
The above performance, by the Bottle Top Boogie Band, was recorded live on Halloween 2013 at The Dolphin (Wellington, Somerset, UK).
StusBlues is close, but it’s Holy Spirit on drums, Jesus on bass, and God on piano. There is no one lead singer. They’re a band, for Heaven’s sake (literally). The music features killer three part harmonies and throwing lead vocals back and forth.
While I cannot agree with your instrument choices, I think you may have inadvertently suggested the bands name:
For Heaven’s Sake
wouldn’t they all be able to play any instrument, including ones not invented yet?
also the light show and special effects too.
I see them as a classic Sixties power trio, like Cream or the Jimi Hendrix Experience.
Or Rush- hey, Jesus could sing, play keyboards and work the bass pedals with his feet simultaneously, just like Geddy Lee.
And Yahweh, of course, was the original Guitar God.
I actually had in mind The Welcome to the Fold Five.
And Mary Magdalene’s the groupie.
Presumably the apostles are the road crew.
Well, of course there would be a Battle of the Bands, with the Trinity taking on Satan and His Helpers, but they are separate bands! It would be a halluva battle, though.
Pops on the organ as big as the universe, Ghost on the glass harmonica and The Dude on acoustic guitar.
If they ever did the 1812 Overture, I can see thunderclaps for the cannon accents. Awesome.
I just don’t see Jesus on lead guitar. I’ll say vocals and tamborine (see Davy Jones).
Spirit is back up vocals.
When on tour special guest Kali and Shiva are the back-up dancers. With Vince Clortho on keyboard and Zuul as the bouncer.
The Buddda runs lights, effects, and the soundboard.
Some contest; we already know the Devil is, at best, just “pretty good” with a fiddle.
the Father is on drums
Jesus is on bass and vocals
The Holy Spirit is on guitar
So basically its Rush.