As for the shower-at-work issue....that varies a lot person to person. I'm in Georgia - so in the summer my morning commute is often already 80 degrees and 90% humidity. I'm lucky because although I get sweaty I don't really smell as long as I change clothes. I think this is because I bike a lot and sweat a lot at other times, so my body has more opportunities to flush the chemicals that make you stink. Now, after riding home in 100* heat, that's a different matter, but then I'm home and hop in the shower. Another good option if you can't get to a shower is a pack of baby wipes kept at your office.
I usually wear baggy cycling shorts and a t-shirt riding to work. I pack undies, khakis, a polo, and socks to wear for the work day, as well as lunch usually. I leave a pair of shoes and a belt in my office, which cuts down on bulk for the daily ride. I ride with clipless pedals and mountain bike shoes when it's cool but switch to flat pedals and sandals when it warms up.
I ride a Trek 920, so it's a rigid mountain bike with road handlebars and racks. I've got custom bags for it because I use it bike camping too, but my main commuter bag is a trunk rack from Jandd. It's mostly water proof , has several compartments, and is expandable in case I need to pick up something small on the way home. Make sure you carry a lock of some sort. I use a cheap metal security cable and padlock because bike theft is rare here, luckily. I don't run fenders because if it's raining in the morning I'm not riding. If I get rained on riding home, that's not that big a deal. I'm usually soaking wet from sweat anyway. Good lights, reflective accents, and hi vis colors are always a good idea, and of course a helmet is mandatory. Here's a shot of the bike in general commuter trim without the Jandd bag. I was out on a night ride with the local bike shop crew a couple years ago.
I went with a beefier mountain bike because I'm riding sidewalks, cutting through undeveloped lots, and cutting behind shopping centers on a daily basis. (When the small knob off-road tires wear out I'll be putting un something like a Scwalbe Marathon. Even riding some dirt, there's no real need for knobs.) This is a southern, suburban military town and has exactly 0 cycling friendly infrastructure. The key to commuting as safely as possible is to think about how traffic flows and where drivers are naturally looking or not looking in traffic, and then use that to stay out of their way. Most of my commute I ride like a punk 12 year old kid, but there are times it's safer to take the lane and ride like a legal cyclist. Knowing which to do to maximize your safety takes thought, experience, and patience. And you still will never be truly safe.
I've got a friend who cycles in SLC, but for sport, not commuting. Seems like there's a pretty strong cycling community out there so at least motorists are used to seeing cyclists occasionally.