Some great advice already..
One thing I find helps is to record the lead vocal, then when you are ready to record a harmony, mute everything except the lead vocal and, say rhythm guitar, something to keep time and tune. Turn the lead vocal down a touch, so you can clearly hear it still but are not competing with it. Once recorded, listen back to the newly recorded harmony on its own, you'll soon hear if it sounds ok. Record each harmony like this, with the others muted, always using the lead vocal as the guide.
Hope that helps