My question to you guys is whether or not I should become really proficient with the bass in EADG before exploring alternate tunings or if I should just start off with what I want to do which is play a bass strung BEAD.
I would say start with BEAD, if that is what you want and you're comfortable with it. I tend to agree with @jbhaugh, actually playing the thing is more important than the tuning. A major scale is still a major, regardless how you string your bass, BEAD or EADG. Sure, you might have to use a different position on the neck to get some of the higher notes, but if you learn your neck well, and location of every note, all that theory applies same as EADG. Again, just different positions on the neck, and it sounds like you might have a local teacher that wants to help you learn from a BEAD starting point. Excellent.
No matter what turning you use I recommend staying in 4ths - the relationships between the notes (in other words, the patterns made by scales) is something you want to keep consistent. EADG, BEAD, EbAbDbGb, DGCF, GCFBb (yeah, really), once you find one note you know where they all are. Picking up a bass tuned DADG, or DAbDAb (yeah, really) can mess with your head.
^^ That.
