A lot of it is just memorization, I've been birding since I could walk. One of the best ways is to sort of "humanize" the songs. For example; hooded warbler sings "tawit tawit tawiteeoo" and ovenbird sings "teacher teacher teacher teacher", black-throated blue warbler sings "I am so lazzyyy", black throated green sings "zee zee zoo zoo zeee" blue-winged warbler sings "Beee buzzzzzzzzzzzz" , Prothonorary warbler is a loud emphatic "sweet sweet sweet sweet!" (always a swampy wooded habitat), Yellow warbler is a quick "sweet sweet sweeter than sweet", chestnut-sided warbler sings "pleased, pleased, pleased to MEETCHA."
http://www.allaboutbirds.org/guide/Chestnut-sided_Warbler/sounds etc.
I also recommend some of the birding by ear cds. They can help immensely. Once you've narrowed down the song to a few birds you can use process of elimination to narrow it down further. For instance, if I hear what I thought was either a junco, pine warbler, worm-eating warbler, or chipping sparrow (all sound very similar) I'd first start with the fact that juncos rarely spend the spring here. I'd look at the habitat the song is coming from. If it's a pine-oak woods patch I'd rule out chipping sparrow as they tend to stick to slightly more open areas. So now I'm left with pine warbler or worm-eating warbler. I can't separate them by habitat as they both favor similar conditions so I'll go back to the song. Worm-eating should have a dry rattle where as a pine warbler is more of a warble/trill with a slightly "sweet/wet" quality to it.
Hope that helps some.