Simplest circuit for making Christmas lights flash
Unfortunately, without knowing exactly what type of product you bought, and having a reliable schematic for it, there's not much advice that can be offered.

As @Brian says, since you're just getting started, you should start with something with less energy than mains.

I'd recommend getting a breadboard, LED, battery, and some components according to any number of online searches for "simple LED blink circuit."

Some things I recommend you research:


powering LED's, forward voltage, and how much current to supply according to the datasheet
current limiting resistors
Ohm's law
resistor-capacitor timing circuits
the venerable 555 timer integrated circuit


If you've gotten to the point where you can blink an LED on a breadboard, you'll be in a better position to ask about making a commercial product modification, but I'd still not recommend doing anything on the mains (AC) side.

Adding or modifying components on the DC side is more approachable and safer, but please seek some help from someone with more experience before tackling anything that uses mains.