Again, welcome to B&B!
Let me start by telling you that the Burt's Bee's shave soap is very difficult, to say the least, for a beginner. It uses no fats (kind of a "thing" with BB products, being all "natural" and stuff), so is fairly difficult to get a proper lather. Col. Conk shave soap (one is Bay Rum since you fancy that scent) is very cost effective and produces a great lather which is good for a beginner. Once you get the hang of proper lather building, then expand into some of the more pricey "boutique" type soaps and creams, or venture back to BB shave soap if that's what blows your skirt up.
As others have said, your razor/blade combo is a very personal thing. What works for one person may not for another. That said, Merkur makes a great line of DE razors, from the "Classic" at one end to the "Vision 2000" on the other. None really produces a better shave than the other, only how it is used and "feels" to the user. I personally recommend the Merkur Progress for a beginner because it's adjustable, giving you some "growing room". Another is a Gillette Fatboy available from eBay. Once you get good at your technique, you can get a more aggressive razor, like the Merkur Slant, Vision 2000, or even a Gillette red-tip Superspeed from eBay
Visit WestCoastShaving.com for some great blade sampler packs. Once you settle with a type razor, you can experiment with blade manufacturers to see what suits your style. Most folks here will tell you that Personna Israeli Red's or Derby Extra's are very forgiving for a beginner because they are comfortable and very forgiving, yet effective at whisker whacking.
Lastly, be sure to read the tutorials. There are mounds of information from the massive amounts of experience available here. I've been DE/Wet shaving for about 8 years and still learned a ton when I joined!
Most of all, have fun! Shaving won't be a chore anymore, it'll become a ritual!
"And thou, son of man, take thee a sharp knife, take thee a barber's razor, and cause it to pass upon thine head and upon thy beard" - Ezekiel 5:1