They are individual( I took the bulbs off of string lights; don't know if it makes a difference)
A "normal" string of christmas lights is wired either in series or parallel, so all of the lights in the string have to be switched on/off (or dimmed) together
. The "old" 7 watt bulbs were wired in parallel, so the internal blinking mechanism could blink the lights individually. But, they have to be rewired if you want use something else (such as the Arduino) to control them individually. The newer miniature (non-LED) light strings are wired in series. There is one "master" blinking bulb, and it blinks the whole string together.
.. and I was wanting to power them directly from the Arduino, so I am not using an adapter or AC power.
Each Arduino output can drive one
regular LED, or a tiny 5V incandescant bulb if it's less than 40mA (less than 0.2 Watts @ 5V). If you want any more "power", you'll need additional circuitry.
Would it be any simpler to go and buy a string of LED Christmas lights and cut them up to use the LED's individually?
the LED lights contain "regular" LEDs that run from about 2-3V @ 20mA. In that case, each Arduino could light one LED Christmas lamp, and you'd just need a current-limiting resistor like any normal LED. But, you'd probably have to do some testing to confirm that because I don't know and different strings could be designed differently.