Advice for LED project
The inexpensive simple solution is to use blinking RGB LEDs.  

Something like this  RGB Full Multi color Flashing LEDs 

No external components required for blinking or color selection. Some of these blinking RGB LEDs randomize the color for each blink, not sure about these.  There are more links to others on this page.





Batteries

You cannot just look at the nominal voltage of a battery to power LEDs. The cuttoff voltage  (i.e. fully discharged) is just as important. 

A Li-ion 18650 cells are generally the preferred battery for LEDs.  The voltage is close to LED Vf and the discharge curve is flat  keeping intensity somewhat constant.

AA batteries operate from 1.5v to 0.8V.  The discharge curve is steep.  This means the LED will get much dimmer as the battery discharges.


Source: ENERGIZER E91 AA Datasheet



Li-ion 18650 batteries operate from 4.0v to 3.0V.  The discharge curve is flat.  This means the LED will not get as dim as the battery discharges.


Panasonic Li-ion NCR18650PF Datasheet





NiMH operate from 1.3v to 1.2V.  NiMH has a very flat discharge curve better than all others.  NiMH has less capacity than Li-ion 18650 and may not last as many hours. 


Source: NICKEL METAL HYDRIDE Panasonic HHR120AA






There is more detail in this answer: 
How to measure Alkaline battery lifetime/capacity in practice LED circuit?

Here is the characteristics of common batteries 
type=>full charge-discharged, volts difference (lower is better), % difference over full charge (lower is better), typical capacity (higher is better)

9V Alkaline =&gt; 9V-5V,     4V,   44%,  300 mAH  
CR123A      =&gt; 3V-2V,     1V,   33%, 1500 mAH
AA Alkaline =&gt; 1.2V-0.8V, 0.4V, 33%, 2800 mAH
Li-ion      =&gt; 3.6V-3.2V  0.4V, 11%, 3000 mAH
NiMH        =&gt; 1.3V-1.2V, 0.1V,  7%, 2300 mAH


If using current limiting resistors use the mid-point voltage of the discharge curve to calculate the value.

Compare the discharge curves 
1.2V NiMH Panasonic NICKEL  METAL  HYDRIDE HANDBOOK
18650 LI-ion battery Panasonic Li-ion NCR18650PF 
9V alkaline Energizer 9V Alkaline Battery
1.2V alkaline Energizer AA Alkaline
3V (not recommended) Energizer CR123A