Don't think in terms of transparency but in terms of masks
Most of the confusion comes from the idea that, if there is "transparency" in the image, there has to be a node for transparency involved, and somehow the alpha information has to be connected to it.
Please read further.
The basic workflow:
1. Determine what part of the object will use the texture.
In Edit Mode select the faces where you want the image to be mapped.
UV coordinates are used to map the image texture precisely on the object.
Press U to Unwrap them (depending on the shape you might need different a projection, in this example I used "sphere projection" but for your mesh it might be different, this is not a post on UV mapping).
2. Create the material(s) and use the image to control the mix of different parameters.
Create a new material and assign it to your object.
To edit the material, open the Node Editor window in shader and object mode.
For blender 2.8 use the shader editor.
Before you do anything else: let's go back the key concept:
There is nothing "transparent" in an image.
An image with an alpha channel, is composed of two different elements. The RGB (Red, Green and Blue channels, or color information), and a black and white channel called "Alpha" channel. White parts determine what is opaque, and black areas what most people call "transparent".
All of this is easier to understand if you don't think at all in terms of transparency and think in terms of Masks, and use those masks to control how elements mix.
The alpha channel of a given image can be used as a Mask, to control the mix of color or the mix of shaders
White areas will be one element of the mix, Black areas will be the other, and any shade of gray will be a partial mix.
1: To control the color of a shader.
If you want to use the color information of the image over the color of a given shader, in a way that all other properties of the shader are homogeneous, then use a color mix node, and have the mask be the factor for the color of the shader. Plug Alpha to the Fac of the mixRGB and the resulting mix to the shader's color input.
Pay special attention to the order in which the sockets are connected.
The black areas of the mask(or "alpha" channel) will use the color set in the RGB mix node, and the white parts will use the color of the image used as texture.
If you are using more than one texture, then use both inputs of the color mix node to mix the textures.
2: To control the the mix of different shaders.
Use this setup if you want to use the mask to control the mix between two different shaders. Note that doing this will make the masked areas have different characteristics.
The mask (alpha) on the image will to control how a shader that uses the colors of your image texture combines with another shader.
Let's see how to deal with transparency then.
Adding an image on a glass object
Glass is not different than any other shader.
If the color image is intended to be part of the glass shader, then use a mix color node. The image's color will be part of the shader's color only in the areas determined by the image mask.
If the image is supposed to be opaque, or be a completely different material, then the alpha mask should control the mix of two different shaders.
And now for
If the image's mask is used to combine a transparent shader with any other shader, then you can see right through the mesh.
Or the transparency can take the color of the image:
- What about the Principled BSDF shader?
It is the same as any other shader. Use the mask to combine colors, shaders or any of the other qualities.
In newer versions of blender the Principled shader has an input for Alpha. If you plug in the alpha channel of the image directly to the alpha input of the shader, you will get a fully transparent object, except in the areas controlled by the alpha channel. In this case you don't need an additional transparent shader.
And, just for fun, you can use the alpha mask in as many combinations as you can think of.
For multiple images with alpha channel overlaid on top of each other:
When working with shaders remember that, RGB information and Masks(based on alpha channels) follow separate paths.
So let's break this in two parts.
First Combine the RGB information using the alpha channel of the image being overlaid as a mask (fac) to control mix over the background image.
Then you can use the sum of all of the alpha channels as a mask to control the mix of color or shaders. You can use a Convert>Math node and set it to add mode.
To mix the colors for a shader color:
For a mix with other shaders: