First off you will want to find your main background image it should be something like the one am using below
Create FOUR new layers (label Rain 1, Rain 2 etc etc), these will be your rain layers (you can use more than 4 if you wish but for this I used four) also remember to - Image > Apply Image for EACH of the four layers!
So take your first rain layer (hide the other 3 for now) then go to Filter > Noise > Add Noise. Use the settings I have used below.
Let's add the Motion Blur filter to the first rain layer also - Filter > Blur > Motion Blur.
Use the following settings.
Now do apply the two filters to each of the other THREE RAIN LAYERS.
Remember for each RAIN LAYER differentiate the Noise & Blur settings!
Once all FOUR RAIN LAYERS are done set them to - Soft Light and an Opacity of 70%. Also that all FOUR are above the background image.
Now we'll add a text layer, clump it into a group this will make the effect look better overall. Colour HAS to be white. Call this layer "Text". Now duplicate the text layer you just created and apply a Outer Glow filter (glow must be white). Set the Mode to Normal - Opacity: 100% and the Size of the glow to 1-3. You may add a Stroke if you wish but I didn't bother. Call this duplicated layer "Lightning Text"
We're onto creating the lightning bolts. So create four new layers and label them Bolt 1, Bolt 2 etc etc. You will need a 1px white brush. Draw a few bolts on each layer at different positions near the text layer, for a better effect start drawing them behind the text. Once you have all four layers done apply a Outer Glow filter to them, same as the Lightning Text Layer - Mode: Normal - Opacity: 100% - Size: 1. If all is done correctly you should have something like this.
(One Rain Layer - One Lightning Bolt & Text Layer Showing)

Finally! We are ready to animate everything.
So switch to the animation window - Window > Animation.
Your first frame should ONLY have the following showing - Text, Rain 1 & Background
Click the Duplicate Frame Button - 
In this frame make ONLY the following show - Text, Rain 2, Background
Duplicate the frame again and have ONLY the following showing - Text, Rain 3, Background
Duplicate once more and have ONLY the following showing - Lightning Text, Text, Rain 4, Background, Bolt 1
You should now have something like below.

Duplicate the frame and have ONLY the following show - Text, Rain 4, Background
Duplicate it 4 more times and have the following show -
Frame 6 - Text, Rain 1, Background
Frame 7 - Text, Rain 2, Background
Frame 8 - Text, Rain 3, Background
Frame 9 - Text, Rain 4, Background
Now for Frame 10 (duplicate frame 9) and have ONLY the following show - Lightning Text, Text, Rain 3, Bolt 2, Background

You should see the pattern am trying to do here so if your clever enough repeat it 2-3 more times.
I suggest adding a couple frames inbetween each "lightning strike" just make sure to change the Rain Layers round each time.
You have reached the end of this tutorial, while it may not be extremly detailed I hope it was enough to help you understand how to create simple rain/lightning animations that you can use in the future.