Question about adding vocal harmonies?

"I wanted to ask everyone, what are the best harmonies? Should I approach them like I do doing guitar solo harmonies?"
 
The best harmonies are the ones that sound best to your ears.  Simple, but true.  You have to be the final judge.
 

I agree with Lynn. I don't think most singers think in 3rds, 5ths, etc. (I'm not even sure what all of those sound like). You can probably hear what you want to do in your head. Getting it out may be the tricky part. Try singing harmonies behind melodies and recording what you do. You might discover that you have an intuitive sense of what sounds good and what doesn't. Personally, I think some of the best sounding stuff happens by accident (or spontaneously). 

In recording harmonies..... avoid what might seem to be the easy path... using something like Melodyne Editor (ME) to make the harmony from a lead vocal. The note changes are generally too great of a jump to do them cleanly and leave no strange sounding artifacts behind. Once you get past a full step to a step and a half, you start running into trouble quickly.  That's not to say that you can't do that and pull it off because you can. I have done it on a few tunes but those harmony tracks are well down in the mix to try to cover the evident artifacts.
 
http://soundclick.com/share.cfm?id=12024980  harmony at the PC & chorus created from the vox lead using ME.
 
The better way is to set at the piano or guitar and solo the vocal track and one rhythm guitar. Pull the fader down on the guitar so that you can hear the chord it's playing but you want the vox to be dominant. Now, using the guitar or the piano, play the exact melody the vox is singing. Take this process section by section. I would also recommend setting up the looping function so that it will repeat for you as many times as is needed to nail the parts exactly. Once you have the lead melody exact now, use the guitar or the piano to learn and play a harmony to the lead vox. You learned the lead so that you got the feel for the groove on the harmony since the note values and phrasing should be exact. No notes should be unison or octaves or not compatible with the underlying chord.  For example if the underlying chord is a C major, you should probably not be singing  a D# (minor 3rd) in it. 
 
Do not limit yourself to only the 3rd, 4th and 5th.....for sure, start there but..... it's even better if you can work in augmented and diminished vocal notes..remember that a harmony vocal is simply using the vocals to create a chord.... for a really really good example of the more advanced harmonies, listen to some of the Eagle's vocal harmonies..... 7 bridges road is a classic. Crosby Stills Nash & Young also come to mind on harmony examples. Quite a few rock groups are just as competent with vocals as they are with their instrumentation. Queen & Bohemian Rhapsody come to mind.... and yes, Def Lepard have some pretty sweet harmony as well.
 
Using this very method, I was able to get a female singer who said she didn't know how to sing harmony, to learn the harmony notes she needed to sing in a song we were working on in my studio. I would play the piano part, she would copy it.... then we'd record it and move to the next phrase.