How many ohms do I need for LEDs controlled by an shift register?
It depends on following three inputs:


the signal or supply voltage \$V_{s}\$ (i.e. the voltage of your shift register output, e.g. 5V or 3.3V)
the forward voltage \$V_{LED}\$ of your LED (depends on the semiconductor material of the LED and that depends on the color; e.g. 1.3V, 1.8V, 2V, etc.). You get this info from the datasheet of the LED.
the current \$I_{LED}\$ you want to operate the LED with (the max. current is given in the datasheet; of course the signal source must be able to supply that current); e.g. 10mA


Then the resistance must be
\$R = \frac{V_s - V_{LED}}{I_{LED}}\$  

Example: you want to operate a red 1.8V LED with 10mA at a 5V source:
\$R = \frac{5V - 1.8V}{10mA} = \frac{3.2V}{10mA} = 320 \Omega\$