When you increase the tension on a string, how is the standing wave affected? I know that wave velocity is the product of wavelength and frequency, and that velocity is proportional to string tension. Does this mean that if you increase the tension on a string, the wavelength, frequency, and velocity will all increase?
 A: This is something that I always see students getting tripped up with.
The equation $v=f\lambda$ just relates these variables together. It does not tell us, in general, how changing one variable changes another. You need additional information.
For example, the velocity of waves on a string is determined by $$v=\sqrt{\frac{F}{\mu}}$$ where $F$ is the tension in the string and $\mu$ is the linear mass density. The wavelength and frequency of the wave doesn't determine the speed. It's a property of the string. Also note that your statement that the velocity is proportional to the tension is incorrect.
If we are talking about generating waves on a string by wiggling one end, then the frequency is determined by how you wiggle the string. Then you can determine the wavelength of these waves by $v=f\lambda$
However, if we are talking about standing waves where only certain wavelengths are allowed, then we can determine the frequency we need to wiggle the string at by $v=f\lambda$
So, your question 

Does this mean that if you increase the tension on a string, the wavelength, frequency, and velocity will all increase?

does not have an answer unless you specify what system you are looking at (although the velocity increases for sure).
A: Think of it like a guitar string, being tuned up to pitch: if you increase the string tension, the string's natural frequency will increase but since the string's length did not change, the wavelength of the string's natural frequency will remain the same. For a standing wave, increasing the pitch while holding the string length the same means that the portions of the string that are in motion will be moving faster. 
