First thing's first: Get the two sprites that you want to palettes-swap, and open them up in MS Paint. You can find all the male front Pokémon sprites here. For this tutorial I've grabbed myself a Bulbasaur and a Squirtle. Place them close together and zoom in 8x.

Next up, selecting the colors. Using the dropper tool, left click on the lightest shade of the color that you want to replace; for example, I have left clicked the lightest shade of green on Bulbasaur.

  Then, using the dropper tool again, right click the color that you want to replace with; for example, I have right clicked the lightest shade of blue on Squirtle.

Then select the eraser tool. This is your key tool to easy palette swapping. MS Paint has a nifty little feature that is: When you right click with the eraser selected, it replaces only the left-clicked color with the right-clicked one, nothing else. Using this, and the colors previously selected, right click and drag the eraser all over your first sprite, being sure to do the whole sprite. The result should look something like this:

Now do the same with the darker shades. Use the dropper tool to select the shade of color that is one shade below the highest. It's usually right next to the highest shade. Remember to left click the one you want to change, and right click the one you want to change it to.

Then you're using the eraser tool again to change the color. Just like in step 3, right click and drag your eraser over the whole sprite. Afterwards, you should get something like this:

Continue to finish every shade of that color. As you can see on my Bulbasaur, there's no green left on it's bulb.

Now that you've done all of one color, move on to the next! Some sprites only have one color (i.e. Ditto), so you won't always have to do another one. Doing the same as before, use the dropper to select the colors and the eraser to apply them.

If you want to, you can swap around and do the other sprite! As always, use the dropper to pick the colors and the eraser to place them. Your final product should look something like this: