First of all, learn to fingerpick, it's incredibly useful.
Second, a great song to learn is Requiem. There's a powertab of it at powertabs.net. Great easy song, and fun too.
The Night and the Silent water has very easy riffs, and you won't have to memorize too much. Plus knowing the opening riff is good enough, that thing owns.
Learn the solo on To Bid You Farewell. Play it without effects first and then add them in. It's a beautiful solo.
If you've got an acoustic guitar and a friend who plays too you might as well learn all of To Bid You Farewell, it's quite easy. Better yet, get a friend with an acoustic, and run your electric through an acoustic simulator, a wah pedal, and a distortion pedal (or get a stomp box and set those prefixes on each pedal), and you'll be able to play the whole song. Fun fun.