Every mode has some important tones unique to the mode If you take the three major modes Ionian, Lydian and Mixolydian, and the three minor modes Aeolian, Dorian and Phrygian as two groups, you could say that in the first major group Lydian and Mixolydian have their own unique tones differently from the Ionian. And the minor group has Dorian and Phrygian with their own unique tones differently from the Aeonian.
So the Lydian differs from the Ionian in having the #11. The other important tone is the major 3rd which gives it the major sound.
The Mixolydian differs from the Ionian because it has the b7. It's major 3rd also important because of the major sound.
Dorian has major sixth in contrast to the flatten sixth the aeolian has. That and the minor 3rd gives it the quality of dorian.
Phrygian has a b9 (minor second) That and the minor 3rd are the important tones. The minor third giving the minor quality also.
Now play with no more than two chords in each mode, the One chord being the tonic, and the other chord having the special tone within it. Examples:
Lydian: C / D / C / D
Mixolydian: C / Gm / C / Gm
Dorian: Cm / F / Cm / F
Phrygian: Cm / Db / Cm / Db
The Ionian mode is more difficult in the sense that it has more development as the tonal Major scale so to speak. Neverthelss I guess you can play with C / F
As the 4th is the important tone (also with the major third and major seventh)
The Aeonian is generally refered to when playing the bVII chord instead of the V chord as the dominant chord. That means basically you can play: Cm / Bb
But also you can play Cm / Ab Because the b6 is the important tone besides the minor third.
Melodically you could hit the strong beats with that special note. The chords I wrote were chosen on purpose, because within them there is the special note each mode needs. You could replace that chord (the ones which go along the C chords) with their respective relative minor or relative major. And generally don't use dominant 7th or diminished because the strong tritone feeling will likely try to push you to a tonal center.
I give you these examples also:
Lydian: C / Bm7
Mixolydian: C / Bbmaj7
Dorian: Cm / Dm
Phrygian: Cm / Bbm7
Ionian: C / Dm7
Aeolian: Cm / Gm7 or Cm / Fm7
You see? Similar to the above ones, but replacing each chord (other than the C's) with their relative minor and major's.
Modal music is like that. Simple few chords.
Now the real interesting stuff is when you do Modal Exchange, and that is basically mixing all the modes (and even the ones which come from the Harmonic and melodic minor scales -or even more which I will not explain that right now) with the same "tonic" or repose.
Modal exchange from Ionian to Dorian (or simply Dorian): Cmaj7 / Ebmaj7(#11)
Phrygian exchange: Cmaj7 / Dbmaj7(#11)
Lydian: Cmaj7 / D7
Mixolydian: Cmaj7 / Gm7
Aeolian: Cmaj7 / Abmaj7
You need to give these meaning, All the Cmaj7 should sound like your tonic, and all the others should sound like a modal modulation. I say this because it is very easy specially with Lydian that if you come back to C, then it starts to sound like you are simply in C Lydian. But it should not be. You have to stablish first your C chord as your Iº of major scale, then present the D7 and maybe you can modulate to the parent scale (Gmaj7) But the momento you change your scale from C Ionian to Lydian (when D7 is presented) that is where the modal exchange comes into play. The others are easier to understand if you are coming and going with those chords. But note this is more intended as a one way road. That is way I explained this last a bit further.