So I've been pointed here. I'm assuming you're working on an acoustic.
Barre chords take a while to get down. It is all about finger strength; Fly's exercise should help, but even without it all you need to do is keep practice. You'll naturally develop the finger strength you need as you go. If you're impatient, you can definitely follow Fly's advice (which, as usual, is spot on) or you can grab a grip trainer and use that. The Gripmaster ones are best for musicians, since they work each finger individually and won't allow your pinky to slack off.
In terms of songs, I think what you're looking for in particular is a category that I tend to think of as three chord songs; songs that are literally just three chords in open position. A few good examples:
Bob Seger - Night Moves
Tom Petty - Free Fallin'
Creedence Clearwater Revival - Bad Moon Rising
Barenaked Ladies - What A Good Boy
Barenaked Ladies - Be My Yoko Ono (actually four, but close enough)
You'll need a capo to play some of those in the same key as the original version.
CCR also does Down On the Corner, which has a riff through the verses and chords in the chorus (specifically F, G and C). The Beatles have a bunch of good stuff too, as Baraka_Guru pointed out. A lot of their early material was based on a 12 bar blues progression, and if you learn that a lot of it will become available to you (along with a bunch of Elvis, Chuck Berry and a lot of other artists from that general timeframe).
If you want a song specifically by a female singer, you can look at Me and Bobby McGee by Janis Joplin. (which was a Dylan song originally, incidentally). It's a little more complicated in that there's a few different chord variations, including seventh chords. It's not terribly complicated anyway and the strumming is easy as all get out.
I can't recommend songs specifically for you to sing, because that will depend on your voice as much as anything else. Regardless, there ought to be something here to get you started.
I may come back with more when I haven't just worked a late shift.
__________________
I wake up in the morning more tired than before I slept
I get through cryin' and I'm sadder than before I wept
I get through thinkin' now, and the thoughts have left my head
I get through speakin' and I can't remember, not a word that I said
- Ben Harper, Show Me A Little Shame
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