Yes, it is possible, but it's harder than making one LED light up, and then doing the other 29 the same.
An LED typically draws 20mA, which is within the limit for an Arduino pin (40mA); however, there is also a 200mA limit for all pins together, so you cannot power more than 10 LEDs off the same arduino. Also, the Arduino only has 13 digital pins, and 5 analog pins, a total of 18 pins. This includes 2 pins used for programming your Arduino.
If you want to run lots of LEDs, the easiest way is running it through an external chip. You can buy a pre-made board e.g. https://www.adafruit.com/products/1429 (24 channel, these can be daisy-chained together for more than 24 LEDs).
Alternatively, you can save yourself some money if you can just buy the chip - a DIP chip, or through-hole, will slot directly into a solderless breadboard, and can be soldered onto veroboard later to make your project more permanent.
There are a number of chips that will do the job, you want an "LED PWM driver" chip - for example, the TLC5940. There is a tutorial for this at http://tronixstuff.com/2013/10/21/tutorial-arduino-tlc5940-led-driver-ic/ - PWM just means you can control the brightness of each LED (individually).
If you want to make your project more permanent, you can solder a board of your own - find someone who can solder (your local makerspace/hackspace will be glad to help, I'm sure!) to teach you the basics, don't burn yourself. The easiest way is to use stripboard (aka Veroboard), this also means you can solder the resistors on to the same board (the LEDs too, if you like!). It sounds very intimidating, but it's not hard to learn. You can then buy PWM chips (make sure you get through-hole, aka "DIP" chips - https://en.wikipedia.org/wiki/Dual_in-line_package ) rather than "surface mount" chips. The chips are sensitive to heat, so to start you might want to solder on sockets (see same wiki page), then you can slot the chips in later.
You should consider very carefully your power supply - find out how many milliamps (or amps) it can provide - every (lit) LED will draw 20 milliAmps, and the Arduino is rated for 200mA, so if you want more than 9 LEDs, you will want an external power source, with an appropriate rating - 500mA = 0.5 amp will drive 24 LEDs. In this case, do NOT use the 2.1mm jack on the Arduino to power it; the transformer will overheat.
P.S. - you will be able to control the exact brightness of each LED independently. Also, if you use RGB leds, and use 3 pins on the IC to drive each LED, then you will be able to control the exact colour of each LED too!