Hi, look at some existing relay boards that you could definitely drive from a MEGA:
Like these: http://goo.gl/3mXu0
Look at the schematic for one here: http://goo.gl/Nnt9g
These only need 2 to 3 ma per pin for the Arduino to "sink". They include an LED per channel so you can test easily.
NOTE: These are "Active low" controls, for good reasons. See the ArduinoInfo WIKI: http://arduinoinfo.info (Relay info here: http://arduino-info.wikispaces.com/ArduinoPower#4-8)
Information on Arduino Pin Issues here: http://arduino-info.wikispaces.com/ArduinoPinCurrent
The seconds display will change 86400 times a day. If this is going to run 24-7 for a long time, you may want to use Solid State Relays for the seconds digits (or at least the 1-second digit). Example here (but you could use smaller ones): http://goo.gl/QhgNO
Can you point to the lamps you will be using? Where can you get them?? Sounds cool...
Let us know who you are doing, feel free to ask more questions...
DISCLAIMER: Mentioned stuff from my own shop...