With a mouth used to being pursed around a reed and blowing, you probably have enough of a clamping ability to use something a bit stronger than a 2 strength, although (as noted above) all of the grading systems differ quite a bit.
If I was bringing you along, I would establish a benchmark with a known mouthpiece (not based so much on quality as just something commonly known by the given teacher), and then work the reed choices from that standpoint. There are mouthpiece equivalence charts around (Woodwind & Brasswind has one on line, I think), and it would be easier for some of us to give advice if we knew what the B & H mouthpiece was compared to others that are more common (at least over here, where B & H equipment is rather thin on the ground.
Rather than buying reeds by the box-ful at first, purchase a "bracket" of strengths in the given brand (i.e., one #2, one #2 1/2, one #3, and so forth), then find out what works with the benchmark mouthpiece. Once you get it refined down and are comfortable with a given mouthpiece and reed strength, then it's time to start comparing the merits of cane versus synthetic, and of individual brands.
(I've made a powerful effort to get along with Legere reeds, having purchased a spread of them for bass clarinet, clarinet and baritone sax. However, the things and I just do not get along. They "fade" on me very quickly, and the edges of the reed have a tendency to tear up my lip tissue. Your mileage may vary, of course.)
One additional caution here, though. Make sure that you get a few solid hours of time with a trained clarinetist on an occasional basis, not some saxophone guy who "picked it up" on his own. I have worked with any number of "self taught" clarinet players who aren't even aware of some of the most basic clarinet technique.
This is not a slam on those who are self taught - the same thing applies in my case to crossing over to saxophone. (Being self-taught, I still don't "get" the bis key.) It's just that playing the clarinet is a different sort of craft compared to playing the saxophone or bassoon, and a lot of the schtick on a particular horn is best learned by emulation, rather than self-exploration.