74HC595 (thruhole or SMD) has 70mA limit on Vcc and Gnd pin, so you should select current limit resistors that allow 8-9 mA. (8 outputs x 9mA = 72mA).
To select a resistor:
(5V - Vf)/.008 = resistor, with Vf the forward voltage of the LED (example, ~ 2.5V for a typical Red LED, some greens and yellows, and often somewhat higher for other colors like blue, white).
(5V - 2.5V)/.008A = 312.5 ohm, so 300 or 330 ohm will do great. 270 would also be okay, for 9.25mA. 1K would cut down the brightness some, but still be plenty bright.
8mA can be quite bright with a modern high efficiency LED.
If you need more current, than TPIC6B595 and TPIC6C595 are controlled the same way as 74HC595 - with clock, data, and latch - but can sink 150ma and 100mA per output pin (shift in a 1, that turns the output on, it goes low to sink current from 5V thru the LED and its resistor. Vs Sourcing current thru the LED/resistor to Gnd).
Don't be afraid to order parts online. Digikey.com and Mouser.com both carry all kinds of parts, and inexpensive USPS mail will get them to you in 2-3 days.
If you want to stock up, spend $20-30 and get a bagful of parts from taydaelectronics.com. You can get a lot of stuff for that much. Parts come from Thailand I think (via Colorado in the US from wht I've received), order a selection of parts that will last quite a few projects.