I learned by doing lots of playalongs and singalongs.
I’d get the guitar tuned to whatever song I was learning, then play rhythm along with the recording until I was comfortable with it. I then sang along with the recording a couple of times, then put the two together. Start with some easy, fun stuff: John Denver is great for this sort of thing.
Eventually, I got good enough to jump straight on the songs without having the recording playing.
This actually lines up pretty nicely with some of the other suggestions that people have posted.
Let’s face it, singing and playing at the same time is like spinning plates. Dividing your attention between two things is always difficult, especially if you are trying to sound musical in the process. When it comes down to it, familiarity is your best friend. The less you are thinking about the guitar part, the easier it is to add vocals. The less you are thinking about the vocals, the easier it is to add guitar.
[As a completely unrelated sidenote, when I was growing up, my first cassette tape (ah yes, the days before CDs) was John Denver. My second cassette tape was Quiet Riot. I offer absolutely no apologies for either.]
Badly.