Lamp/Alarm Clock with Wireless Control - Feasible?

I'm trying to come up with an interesting, long-term project to work on and get some experience doing embedded programming. Ideally, I'll have something cool to play with about six months from now.

Toward that end, the project I envision pretty much consists of throwing some strong LEDs and some kind of wireless receiver on an Arduino board, and using another board with a wireless transmitter and a pushbutton to send on/off signals. I'd like to be able to wirelessly send my own alarm sounds/music to the alarm. I'd also like them both to have their own wireless power supplies.

For the first (control) board, I think that I'll need:
1x Standard 16x2 character LCD - for time setting, song selection
4x buttons for menu navigation, etc. - how will I get these?
1x Arduino Uno - the latest board
1x Wireless Transmitter - http://www.glacialwanderer.com/hobbyrobotics/?p=291
1x Mass storage shield - does it matter what I use here? I mean, between SD and USB.
1x 9v battery adapter - Arduino Playground - 9VBatteryAdapter

For the second (clock/lamp) board, I think that I'll need:
1x Arduino Uno - the latest board
1x Wireless Receiver - http://www.glacialwanderer.com/hobbyrobotics/?p=291
1x Mass storage shield - to store sounds to be played
?x A bunch of very bright LEDs - do very bright LEDs even exist? I want it to work as a lamp on my desk, but would settle for something bright to help me see a little better waking up when its dark
1x 9v battery adapter - Arduino Playground - 9VBatteryAdapter

At this point, I know nothing of circuits that wouldn't be covered in a general, university-level physics class. Is that going to be a big problem? I've also never done any soldering and don't even own a solder - is my inexperience with this sort of thing going to be a problem? As a senior CS student, my background is entirely in software, which kind of worries me.

If anyone has suggestions for a more interesting project, I'd appreciate them very. I'd like to make something I could actually use at some point, rather than leave on my shelf when I'm done with it.

4x buttons for menu navigation, etc. - how will I get these?

Radio Shack? Lots of places have momentary contact push-buttons switches.

1x Mass storage shield - does it matter what I use here? I mean, between SD and USB.

It most definitely does. A USB memory stick is a slave. You need a USB Host device to talk to it. The Arduino is a USB slave device, too. Uh oh. You'd need a USE Host shield, and to write a device driver that fit in the Arduino's programming space. Not a trivial task. Writing to, and reading from, an SD card, on the other hand, is easy.

1x 9v battery adapter - Arduino Playground - 9VBatteryAdapter

Price per watt hour, the 9V battery comes in at the bottom of the list. 4 AA's would be a much better choice.

1x Wireless Transmitter - http://www.glacialwanderer.com/hobbyrobotics/?p=291

Those things work. Barely. There is a lot of interference and lots of garbled data. You'll need a lot of code to get good data transmitted with them.

?x A bunch of very bright LEDs - do very bright LEDs even exist?

Blindingly bright, hurt your eyes, LEDs are available.

At this point, I know nothing of circuits that wouldn't be covered in a general, university-level physics class. Is that going to be a big problem?

Not really. Most of the stuff you have selected is pretty much plug and play (or pray). Plenty of people here do know circuits and circuit design and component choice, etc. if you get to the point where you need that kind of help.

I've also never done any soldering and don't even own a solder

While you are at Radio Shack getting the switches, get a soldering iron and some solder. Practice soldering some wires together. It really isn't difficult, with a clean iron.

As a senior CS student, my background is entirely in software, which kind of worries me.

It's a shame that the hardware side of things is so neglected. That doesn't mean that you can't remedy that.

I'd like to make something I could actually use at some point, rather than leave on my shelf when I'm done with it.

If it interests you enough to complete, and you are proud of it, you'll use it. If not, something else will suggest itself.

You can get a nice 20 button keypad from surplussales.com, search the switches page, about 1/2way down. $1.
You will need 2 arduinos, one to send RF and one to receive it. The transmitter will work great if you base it around an 8MHz Promini and 1000mAH LiPo battery, and virtual wire. Go into powerdown sleep mode after each button press, use the button press as interrupt to wake up.
mpja.com has 10,000 and 15,000mcd SUPERbright white LEDs. These things are so bright I run them at less than 2mA, and I only have 3 hooked up. They also have LED assemblies that you could drive with very little effort. I've seen similar in autoparts store, intended to run from 12V most likely.