All LEDs have a maximum current specification. If you exceed this value, you will most likely destroy the LED immediately. It's probably best to plan to use a current that is 50-75% of the maximum rated current. That way you'll never blow an LED and get more lights to boot for a given batery size.
Different color LEDs have different voltage drops across them for a fixed current. RED LEDS have the lowest voltage drop, and Blue LEDs have the highest. If there were no losses, the voltage drop across a red LED would be 1.2 volts, a green LED would be 1.45 volts and a blue LED would be 1.9 volts. In practice these values are typically ~2 volts for red, ~2.7 volts for green and ~3.5 volts for blue LEDs.
There are lots of ways to connect multiple LEDs. The simplest is a parallel connection of single LED/current limiting resistor strings. If you are using a 9 volt battery, this methods wastes most of the battery power in the current limiting resistors.
For example, let's set the current to 10 ma. from a 9 volt battery. A red LED will have a 2 volt drop so you need to drop 9-2=7 volts at 10 ma through the resistor. V=RI or R = V/I = 7/.01 = 700 ohms. For Green the resistor value would be (9-2.7)/.01 = 630 ohms and for blue (9-3.5)/.01 = 550 ohms. If you wanted just a single red, blue and green LED this method will uses 30 ma.
A better way to go is to put a string of several LEDs in series and limit the string current with a single resistor. A string of a red, green and blue LEDs would require a resistor with the following value.
V rest = 9 - (2 + 2.7 + 3.5) = 9 - 8.2 = .8 volts. R = V/+ = .8 / .01 = 80 ohms. (A slightly greater resistor value such as 100 ohms would be ok since it is easier to get.)
By putting 3 of these three LED strings/resistor chains in parallel, you can power 9 LEDs verses the 3 LEDs in the first example using the same 30 ma of battery current. This method is much more efficient.
There are many other ways to do it even more efficiently but they require multiplexing of a many phase clock. It's not that hard, but you need to do some circuit fabrication.
You can get a lot of good information on LEDs and driving circuits at https://www.maxim-ic.com/Display.cfm