You can cast a song every 8 seconds (6 with fast cast). The song actually takes effect after 6 seconds (4 with fast cast). If you move before that, your song will most likely be interrupted. After the song takes effect, you can move with no ill effects. Whether someone receives the effect of the song depends on their distance to you. There is no visual indicator however, so you will have to eyeball the distance. Also, the song's area of effect is always centered on you. Thus, you will have to do a lot of running.

Positioning is important. This is because each party member can only have two song effects at one time. In the midst of battle, the mages won't want minuet and the meleers won't want learned etude. Ideally, you want your group to be positioned like so:

MOB o - Fighters * - mages + - you

This way, when you need to cast madrigal and minuet, you can move up and only affect the melee group like this

And when you need to cast ballad and etude, you can move back like this

At the end of the battle, if you want to cast paeon and ballad on everyone, switch to a harp (for greater casting range) and move to the middle like so:

A typical battle goes something like this:
1) Run to Melee
2) Sing Madrigal
3) Sing Minuet
4) Sing Requiem on the Mob
5) Cast a bar-ra spell if necessary (and if you're sub WHM)
5) Run back to Mages
6) Sing Ballad
7) Sing a Threnody on the Mob
8) Sing an Etude for your Mages
9) Wait for monster to die (and cure, dispel, whatever)
10) Run up and cast Paeon on everyone.
11) Cast ballad on everyone.

At the end of step 8, you will have 2 defensive songs on your fighters, 2defensive songs on your mages, 2 offensive songs on the mob. Probably takes about a minute to do.

To make things quite a bit more complicated, there are two types of instruments. Wind instruments, like the flute and cornette, and string instruments like the harp. Wind instruments have a tight range, and string instruments have a much larger range. Thus, if you're playing paeon and trying to cover the whole party, you should equip a stringed instrument.
Likewise, if you're playing minuet, you should probably equip a wind instrument. Thankfully, this can be macro'd. My macros are set up like such: