Rhythm is a delicate thing. It's a topic that should be studied with great care on its own. It deserves more time in the spotlight. I've devoted a little time to rhythm guitar in an old post:
The study of rhythm is a great thing to do. Take some lessons with a drummer or percussionist. Listen to lots of different music and try to get a handle on what's going on with the rhythm.
So now you've studied rhythm and want to sing and play your instrument of choice at the same time...Now what?
- Learn each element separately
- Put them together SLOWLY
Right now I'm working on singing and playing banjo at the same time. I know the singing part but I'm having trouble pairing it with the picking. I'm finding that if I put the metronome on slow and work through the parts very slowly,] I can smooth it out with not too much trouble.
Bring the tempo up slow and with ease and eventually it'll be up to the tempo you want.
- Record yourself
- Be open to change/Less could be more or maybe not...
The challenge of singing while playing is a fine art. It offers limitless opportunity for musical creativity. Get good at your instrument. Get good at singing. Listen to the great ones. How did they do it? Steal from them, eventually you'll find your own style through the process.
Hope it helps.