1. The index of refraction is quite often used to determine how much light rays bend upon moving from one material to another. The changing directions effect is a direct consequence of the light changing effective speeds. So the first question is if the problem is about light bending at an interface or is about it changing speeds. If its about changing directions, see Snell's Law of Refraction.

2. Determine if the index of refraction is increasing or decreasing so that you know if the speed is increasing or decreasing.

3. If the problem involves wavelengths and frequencies, you may have to use the equation $v=f \lambda$. You may also have to use the fact that the frequency is constant accross a border, the wavelength and speed are not.

4. Solve for the desired quantity.

• ... treat the wavelength as the constant accross a boundary. Frequency is what is constant accross a boundary.
• ... try and use Snell's Law for problems that only involve speed or wavelength changes. If the problem does not involve the bending of light, you do not need to use Snell's Law.
• ... believe that a light photon actually travels slower than c. In reality if a photon (bundle of light) exists, it travels at c. In a medium there is vacuum between the atoms and that is where the photons exist. The interaction with the atoms in the medium, via mechanisms like scattering, is what makes the light take longer to travel the same overall distance.
• ... read too much into the statement above. You can just treat the light like it's moving slower than c.

• Sketch a simple physical representation where the wavelength of the light changes and the speeds are labeled on each side of the boundary.
• Memorize $v=f \lambda$ and $n=\frac{c}{v_{eff}}$
• Frequency is a property of the source, not the medium. Frequency does not change accross a boundary.
• Index of Refraction related to speed changes is taught breifly before Thin Film Interference so that you can find the wavelength in the film. If it wasn't for TFI, it would be taught in junction with Snell's Law of Refraction where it is often used as well. 
• It's not too hard to use the concept of the speed changing accross a boundary, coupled with the frequency not chaniging, to find the change in the wavelength. That feature, along with speed is distance over time and some geometry, makes deriving Snell's Law as a consequence of the speed changing very doable. Go through that derivation and you will have effectively mastered the concept of the index of refraction (at an introductory physics level).