How to play Pandora (or any music source) throughout your home using wireless speakers

There are of course fully integrated systems to accomplish this (such as Sonos or even getting your home hard-wired). The following is a much cheaper way to play music in any room wirelessly, plus full remote control via your mobile devices.

Disclaimer: we did this in our home over two years ago, so my recollection isn't perfect - please let me know about problems or missing steps and I'll update accordingly.

 Ingredients:

1. Hardware
1. Home Base Computer. Choose a laptop or desktop computer that will act as the home base for your music source. We use a MacMini as our dedicated home base.
2. Wireless Speakers. Purchase a pair for each room/floor in which you want sound. We have several sets of AudioEngine A5 speakers ($399 a pair - link), which are truly excellent speakers specially designed to be combined with Airport Expresses (more on that below). But you can use any wireless speaker - we also use a pair of RocketFish speakers ($129 - link) for a bathroom.  (Note that the term "wireless speaker" means the speakers will connect to a wi-fi signal and therefore do not need to be hardwired to the source of the music - they will of course have a power cable and cables connecting one speaker to its pair.)
3. AirPort Express Base Station. Purchase one ($99 - link). This is a wireless router and may replace your current wireless router (the router is the box that's connected to the modem provided by your cable company, eg. Comcast).
4. AirPort Expresses: Purchase one for each set of speakers. These devices essentially create mini networks to extend your existing wireless network.  It appears that Apple discontinued these in anticipation of a new Airport Express Base Station, but I haven't seen a new release of the Expresses, which are still available on Amazon (used/refurbished, $60 or less - link).
2. Applications
1. Airfoil for Mac/Windows: You should download this application on your home base computer (free to try, $25 for license key, which you will need - link).  This program will allow you to select which application you'll stream the music or sound from (Pandora, Firefox, iTunes, etc.) and also let you turn the music on or off in a given room or adjust volume.
2. LogMeIn for your computer: Download this application on your home base computer (link).  The desktop software is free and you'll have to create an account.  This software will allow you to view and control your home base computer from your mobile device.
3. LogMeIn Ignition for your mobile device. Install LogMeIn Ignition (find it on the mobile app store on your phone) on the mobile device that you want to act as a remote control.  This mobile app costs $29.95, but you can download it for free on multiple devices (must be the same operating system) as long as you're using the same account on each device (eg., iPad, iPhone).  This app communicates with the desktop software to allow you to "screen share" from your mobile device - you will be required to jump through a few hoops to allow your mobile device to control this computer via the LogMeIn Ignition mobile app (such as generating and entering a pin).

    Total cost, assuming 3 sets of AudioEngine speakers: $1,530.95

Instructions:

1. Set up the AirPort Express Base Station. This process assumes that you'll be replacing your existing wireless router with Apple's Base Station; there may be a way to add AirPort Expresses to a non-Apple wireless network, but we're Mac people and I just haven't done that (if you have, let me know!).  You'll be connecting the Base Station to your cable modem (the box your cable company provided) using an Ethernet cord - you should follow the set up instructions provided with the Base Station.  Again, this creates a wireless network around your whole home (which you may already have).
2. Connect the speakers to AirPort Expresses. Set up your speakers in each room/floor of the house you desire. If you're using AudioEngine speakers, simply plug an AirPort Express into the back of the speaker and plug one end of the provided (short) black cable into the bottom of the AirPort Express and the other end into the "audio in" jack in the back of the speaker. If you're using a different wireless speaker, plug the AirPort Express into an outlet near the speaker, then connect the speaker's receiver (usually a separate device that you also plug into an outlet near the speakers) to the AirPort Express by plugging the appropriate cable into the jack at the bottom of the AirPort Express.  The light on the AirPort Expresses may be yellow until you accomplish step 3 below, at which point the light will turn green.  This step creates a connection between your speakers and the applicable AirPort Express.
3. Connect the AirPort Expresses to your wireless network. If you're using a Base Station, this should be a snap - using AirPort Utility (an application provided with the Base Station), you'll be able to add each AirPort Express to your wireless network (assuming they don't connect automatically). Follow the instructions included with the AirPort Expresses. You will likely want to name each AirPort Express mini-network in a way that helps you remember where the speaker is located (eg, "Master Bedroom", "Office").  This step allows the AirPort Expresses (and by extension, the speakers) to talk to your wireless network.
4. Connect your home base computer to your wireless network.  This is the computer that will be the source of all your music, so you'll want Pandora or iTunes on this computer (or whatever music source you desire). Confirm this computer is connected to the wireless network created by the Base Station (open up a browser window). Confirm your music source plays properly out of the internal speakers on this computer.  
5. Airfoil.  You should have Airfoil installed on your home base computer.  Configure it properly as prompted - each of your AirPort Expresses should appear in a list in Airfoil, and the speaker symbol next to each one should be lit "blue" to indicate they are active (click the speaker icon next to each one if not).  From the drop-down menu at the bottom of the Airfoil interface, select the music source you wish to use - you'll first have to select "Other application..." and find Pandora (iTunes, etc.) from the list of applications.  Any program you add will be there the next time automatically.  
6. Test the system.  At this point, you should be able to control the music from your home base computer and have it streamed throughout your house from each speaker set. If one or all speakers aren't playing music, confirm that (a) you've connected the speaker wires properly and the speaker is working independently of the system, (b) you've connected the speakers to the AirPort Expresses properly, (c) each AirPort Express appears in your AirPort Utility application as a functioning network, and the light on the AirPort Expresses is green, and (d) Airfoil is detecting each speaker network, each speaker is active within Airfoil, and you've selected and are playing the correct music source.  
7. Adding remote control capabilities.  The LogMeIn desktop software should already be installed on your home base computer. Open the LogMeIn Ignition app that you've downloaded onto your mobile device. You should be able to select your home base computer and connect to it (this process can take a moment).  If it's functioning properly, you'll see the screen of your home base computer and the movement of your finger will move the mouse on the screen - this can be a bit tiny so you will likely want to zoom in using the magnifying (+) symbol on the menu bar on the bottom of the screen.  (Note that on a Mac, you can also connect to and control the home base computer using Screen Sharing.)  This will allow you to control whatever music source you're playing on the home computer as though you were sitting in front of it. 

Caveats and Tips:

1. Once or twice a year, our wireless speakers develop "spottiness" - the signal periodically winks and sound stops streaming for a few moments. This happens more frequently with our non-AudioEngine wireless speakers.  Try making sure the receivers are near the speakers and the AirPort Express (for example, walls can block the wireless signals).  Other times the AirPort Express signals may be interrupted; it's rarely obvious what the cause is for either of these scenarios.
2. I highly recommend paying the $36 per year for an ad-free Pandora subscription. You won't regret it. However, even with a paid subscription, if you let a station play for hours at a time, Pandora will ask you to confirm whether you're still listening.  Also, a song often gets cached by Pandora - when we wake up in the morning and play Pandora, it will often play a few minutes and then stop. Pandora will report a connection problem and you'll just have to prompt it to retry the connection. I deal with this by skipping to the next song when I open up Pandora in the morning; it will then immediately prompt me to retry the connection, saving me from having to do it in a few minutes.
3. Remember that you can play any sound source using Airfoil - if you want to listen to the public stream on NPR or any public radio station via www.tunein.com, simply add/select your browser as the sound source.