Will the processor by itself have enough power to power all 30 LED's plus any other parts I may need.
No! One Arduino output can power about 1 LED. The spec is 40mA maximum from any one output pin, and I believe 200mA total. (A standard-everyday LED requires about 20mA.)
I thought I may need a Relay, but then, I believe, that I would not be able to pulse the LED's through the relay.
A relay should work fine.
Get a relay with a coil rated at 5V, and a coil current of 40mA or less. (There are some relays especially designed to work woth the Arduino.) The contact will need to handle the LED power (voltage & current) but just about any relay should be able to turn on/off 30 regular LEDs.
If you get a solid state relay look for one that works with a 5V control voltage. They usually work with a range of voltages and 5V solid state relays are far more common than 5V mechanical relays. Make sure the solid-state relay is rated to control DC.
Do you know how the LEDs will be wired? LEDs require a "constant current" power supply, and with standard (low-power) LEDs, that means just a resistor in series with each LED.
I was asked to help a student create a stand of 30 LED lights that pulsed on and off for five minutes at a time.
I assume that means on for 5 minutes and off for 5 minutes? If the timing is not too critical, you can use an [u]NE555[/u]. A 555 timer is less than ond dollar and it's "programmed" by selecting resistor & capacitor values so you don't need any software programming. But, the "analog" components are not as accurate as a crystal (or resonator) in a microcontroller. You could also build a crystal -controlled hardware-timer, but the Arduino and a littel software would be easier if you don't already know how to do it.