Another decent, and cheap, tabbing program is Tabit. It’s so cheap, it’s free.
You can upgrade and register it (I think for like $25), this gives you a few extra doo-dads to it, most importantly the ability to play more than one track at a time. That is, you can tab out your song, layer a bass line on it, add some horns, maybe a solo track, you can even program a drum beat.
It doesn’t come out sounding like studio quality, by any means, but it’s pretty damned nifty.
The site has a ton of transcribed songs too, it’s just that some of them are transcribed by tone deaf people with no sense of time, but most of them are accurate.
This is not a song suggestion, but one random thing that helped me when I was beginning was the realization that the simplest voicings (finger positions for the same chord) were not neccessarily the correct one for a song. For example, when going from F (1st position) to G to F#m (2nd position), the “normal” open position G (320003) is less smooth than the 3rd position G (355433). So, if a particular chord transition is really annoying you, a different voicing may make all the difference.
As for song suggestions:
I second Good Riddance (Time of your Life) because it’s a nice intro to picking patterns through some of the power chords
American Pie by Don Mclean is kind of neat in that it’s mostly just strumming through a decent variety of chords, but there are a couple of spots of quick progressions, if you feel up to tackling them
Tracy Chapman’s Telling Stories uses an oft-heard chord progression (Em C G D) and sounds pretty good with just voice and acoustic guitar
If you can get someone to show you open G tuning, it is very easy to play Norwegian Wood just by playing the lead notes on the two A strings with your pinkie and index finger. I was never better than an awful guitarist but for years my wife thought I was terrific based on hearing me play that one song.