Welcome to the board! I've only been riding a year and a half, and got my "road" bike about 4 months ago. It was definately a progression for me, starting with a Trek cruiser style bike, moving to a Giant commuter bicycle where I was stil upright, but not as much - the Giant is substantially lighter than my Trek, and that took the most time to get used to. First time I rode it at any kind of speed it scared me as it's so much more responsive to any movement I make. Then I got my Fuji in February of this year - a lot lighter bike, but I was already past the weight issue - now it was a different riding position. I was fortunate enough to have a good fit from the beginning with the bike, so I was comfortable physically sitting on it. I had issues with the first pedals I had on it, as they were road pedals and I could not unclip for the life of me- fell over in the parking lot the day I bought it and left to ride home! Changed them out a week later to SPD pedals, and once I was comfortable clipping in and out (with DH holding the bike for me), I started riding on bike paths or on the roads during the weekends or early morning hours when traffic was lowest in order to get used to it. Practice, practice, practice!!!

As for traffic, when I started commuting, it did take a while to get used to it. I'm actually a very aggressive and defensive rider (and driver) - whether on the bike, or behind the wheel, I'm always looking out for the next idiot to cut me off or hit me - probably comes from riding a motorcycle. I make eye contact with drivers, especially crosstraffic who are likely to pull out in front of me - a lot of people don't realize how fast a bicycle is travelling and think they can "make it" across the street ahead of you - or they "see right through" you as they're looking at CAR traffic and don't even realize you're there. I will look at the driver, making eye contact, and especially on a downhill if I'm hauling butt, I'll shake my head "NO" at them, so they know I'm continuing straight in front of them, and that I know I have the right of way. No matter what, I always cover my brakes in case I need to stop, and if I see that they are still "creeping" out in front of me, I'll start to slow down in order to make any kind of evasive maneuver if needed.
I will also "take my lane" when coming up on an intersection so the cars behind me know what my intentions are, and don't try to cut me off to turn right in front of me (either intentionally or not) - especially important when there is a right turn lane, make sure you're not along the curb if you're going straight, but get over in the lane to go straight and make your intentions known!! Likewise, I will signal that I'm changing lanes to that straight lane so traffic knows that's where I'm going.
I also ride with head and tail lights on my commuter (and a tail light on my other two bikes), and turn them on DAY OR NIGHT for more visibility. My commute has quite a few shadowed areas and bridges where a driver's visibility is compromised by pupil constriction - the lights help a lot in making yourself seen.
I guess the point of these things are that by making myself visible, and alerting drivers of my intentions, it makes me "feel" more safe. That gives me peace of mind, and makes me more comfortable.
As to your issue of not seeing vehicles, I have that problem. I have an....attention issue All joking aside, seriously, I sometimes don't see what's right in front of (or behind) me. Not a good thing! Once you're comfortable with the physical act of looking over your shoulder while moving without pulling to that side, you'll be able to compensate by looking more than once or for a longer period of time to make sure a car isn't there. I've also found that some of my glasses actually obstruct my peripheral vision which forces me to turn even further to really look behind me as opposed to simply turning my head a little and looking out of the side of my eye - I hope that makes sense In other words, maybe the car was there all a long, but the arm of your sunglasses blocked it and you couldn't see it until it came closer to you and was in a greater field of your vision. You can also try to make your looks back more useful by looking further back. I try to look down the street, past what's right next to me so I know what's coming. If I'm coming up to an intersection that I need to turn left on, I'll start looking back when I'm like 100 yards or more away from the intersection to see how traffic looks so I know how soon I need to try to get over. That gives me time to adjust my speed in order to get over, I know approximately how many cars I saw back there, and can make my move accordingly.
Lastly, I know a lot of people here do listen to music while riding - I don't. Only if I'm strictly on a separate bike path where there are no cars will I wear an mp3 player. Riding to work in traffic on the street, I keep my ears open as I, like you, also listen for traffic. If I can hear that there's a car coming up behind me, I may not even HAVE to look - I can hear it and know it's there, and probably close enough for me to touch so I wont worry about looking and risk going over into its lane. Use all of your senses especially when you're starting out and aren't as comfortable yet as you will eventually be!!!
It really isn't doing any more than we should be doing when we drive (with the exception of the music!!!) rather it's being more conscience about making sure we're actually doing these things since the threat of physical injury is greater when we aren't surrounded by metal. Just like when we learned to drive the more you do it, the more comfortable you'll be at it. Just get comfy on your bike, practice the movements that you'll need to do on the road off the road first, get visible, then ride, ride, ride