How to get there?
Take Iceland Air. No one else flies direct from the US to Iceland; if you do anything else, you fly past Iceland to the UK or mainland Europe, and fly back. Ugh. For those in the western US, they recently started service from Seattle. Because Iceland is so far north, going to NYC adds a lot of travel time; Seattle-Iceland is 7 hours 15 minutes, NYC-Iceland is 5 hours 45 minutes. They also fly from Minneapolis, Boston, Toronto, and elsewhere.
On your flight from the US to Iceland/Europe, do whatever you can to sleep on the plane, or you'll probably be dead tired when you arrive between 6 and 8 am and will both lose that whole day from being too tired and take longer to shift time zones. I regret not having used alcohol to encourage sleep.
Know in advance whether your hotel is within walking distance of the BSI bus terminal, where the airport shuttle's transfer point is; it's cheaper to go there than to go direct to your hotel. Also, know in advance whether you want to do any tours en route to/from the airport.
When you get there?
Get cash from the airport ATM. Go get a seat on the airport bus or on a tour that leaves from the airport, from the tour desk there. If you're headed straight to town, you can also use the computerised kiosks, which will probably be faster. Go ahead and buy a round trip, unless you think you'll lose the paperwork. Be pleasantly surprised at how well most of the residents speak English.
So, I hear their economy has issues?
Yes. Basically, their banks caused the entire country to go bankrupt. The result is that some stuff is at the usual way-expensive prices (food and wine, most activities targeted specifically at tourists), while other stuff is the same as or cheaper than in the US (not normal, local public transit, groceries). Some stuff is even priced in Euros instead of Icelandic Kroner. This makes it hard to compare prices.
Paying for stuff...use your ATM card to get cash, or use a credit card. In Europe, I tend to get cash for about half of my purchases, using a credit card for the rest, and using the purchases near the end to balance it out. Tourist places are usually kind enough to let you pay cash for some and use a credit card for the rest, to use up your spare cash. I recommend doing this to get rid of all the money you don't want as souvenirs, unless you or a friend will use it in the near future. We spent enough time in Europe that I got a Capital One credit card; unlike most credit cards, they do not add a surcharge for foreign transactions. ATMs are as common in Reykjavík as in the US (as well as most places in Western Europe). Let your bank and/or credit card issuer know in advance when & where you are traveling; otherwise, they will often decline foreign transactions due to concerns of possible fraud.
What to do there?
First off, every global mapping website seems to be screwed up for Reykjavík. Try ja.is, an Icelandic website.
You could spend your entire time in Reykjavík, but I recommend you spend a fair bit of time out of the city; Iceland's coolness lies elsewhere. Both Iceland Excursions and Reykjavík Excursions offer a slew of tours, often the same ones. Several hotels will be closer to the Iceland Excursions office, in which case you should use them. (We did, and liked them.) Reykjavík Excursions runs the major airport shuttle, and also has more tour options that leave direct from the airport (they even take care of your bags!) and drop you off at the hotel, particularly when direct flights from the US arrive.
If you're there mid-September through mid-April, try to see the Northern Lights at night.
If you're only there for a day, as part of a stopover to/from Europe, I recommend you pick two of these three options. Visit Blue Lagoon (direct from the airport); it's a tourist trap spa but well done, a unique experience because it's based around a cluster of natural mineral-filled hot springs and a nice relaxing experience. Wander around Reykjavík for part of the day; there's definitely a half day worth of cool things to see. Go on one of the shortened Golden Circle tours (5.5-6.5 hours instead of the usual 8-8.5) in the afternoon or evening to see Gullfoss and Geysir. (If you do both Blue Lagoon and the shortened Golden Circle, schedule it as a package, they should just transfer you directly from one tour to the other, and they'll tell you which bus to get on leaving the Blue Lagoon to make it work. You also probably want to pack your own lunch, if you don't want to buy food at tourist-trap prices.)
We spent roughly two half days exploring Reykjavík. We wandered along the main shopping street (Bankastræti/Laugavegur), the main arts-store street (Skólavörðustígur), Tjönon (The Pond), and part of the waterfront (along Sæbraut). We went up top the top of the church on Skólavörðuholt (if it's still under construction, go inside but don't pay to go up, the construction scaffolding blocks too much of the views) and to the viewing platform on top of their hot-water reservoirs (Perlan ("The Pearl")). We also paid a visit to one of the city pools; they're all geothermally heated and have nice relaxing hot tubs; many have saunas as well. That said, I recommend you do at least one thing not on this list; I'm sure there's cool stuff we didn't even find out about.
While you're in Iceland, consider trying whale meat. We ate at the Sea Baron, on the north side of Geirsgata, near where Tryggvagata merges with it; they sell fish kabobs, including whale and halibut. They also had scallops, potatoes, and other yummy-looking things. Both the Lonely Planet and our bus tour giude recommended them. There are also more upscale restaurants with whale meat; just ask for recommendations.
If your hotel is near Bankastræti, Skólavörðustígur, or the west part of Laugavegur, there's a fair bit within walking distance; regardless, you may want to use the city bus system. We tried to buy day passes a few different places; in the end, we could only buy them at the Hlemmur city bus terminal (Laugavegur & Rauðarárstígur); they also had a great flyer with a city map, a bus map, and a bus schedule that I hadn't been able to find elsewhere. Know where you're going, and be able to point at it written down, or have it circled on the bus map so you can show the driver for help finding it; one bus driver was the only person we encountered in Iceland who didn't speak English.
If you're there for two days, consider the more leisurely Golden Circle tour. If you're there for four or more days (or maybe even three), look into a multi-day tour with an overnight stop elsewhere, or rent a car or go hiking/backpacking to explore on your own. (Even though we were there for four days, we lost over a day to stuff related to the Reykjavík Marathon, and most of a day to not sleeping on the plane.)
Wikipedia has decent articles on Reykjavík and Iceland, as does Wikitravel (Reykjavík, Iceland).
Last but not least, get a travel guide. If you're only there for one or two days, you'll probably be just fine with a Europe book that happens to cover Iceland (note that Rick Steves does not) as opposed to one that's specific to Iceland. We were quite happy with our Lonely Planet Iceland; it was full of great information.