Finding expected number coin flips to get 2 consecutive heads First, I know what the right answer is, and I know how to solve it. What I'm trying to figure out is why I can't get the following process to work.

The probability that we get 2 consecutive heads with one flip is 0. The probability that we get 2 consecutive heads with 2 flips = 1/4. The probability of getting 2 consecutive heads with 3 flips = 1/6. The probability of getting 2 consecutive heads with 4 flips = 2/10 = 1/5. And the probability of getting 2 consecutive heads with 5 flips = 3/16. 

Am I doing something wrong. I don't see any easy way to use these numbers to solve the original problem of finding the expected number of coin flips to get 2 consecutive heads.
The problem is that your experiment is not well-defined. To solve the mentioned expectation problem, you would define the experiment as follows:

You toss a coin until you get two heads in a row and then you would stop. So there are no two consecutive heads before that.

each of the probabilities, that you have calculated in your process, is related to the following experiment:

you toss a coin n times. what is the probability of having 2 heads in a row.

By the way, according to the last experiment, you have miscalculated the probabilities in your process.
The definition of expected value in this case is:

$$E[x]=\sum_{x=2}^{\infty}xP[x]$$

and

$$P[x]=\frac{x-1}{2^x}$$

Combining these yields:

$$E[x]=\sum_{x=2}^{\infty}\frac{x^2-x}{2^x}$$

Therefore $E[x]=4$.