I went the opposite way, guitar then bass. However, I still play both extensively so have a few things to mention:
1. Definitely do not learn in anything other than standard tuning (EADGBe), which others have already mentioned. Unless you play a 6-string bass (or a E-C five string), the B string won't be an issue anyway.
2. Learn songs you like, really helps the motivation.
3. Start with chords, it will be the most useful addition to your playing/songwriting arsenal (hey, many people only know how to strum chords and still have fun around the campfire

). It will take a while to get comfortable, not sure if your bass playing will speed that learning up or not.
Later
Rob