The dominant technique for expected-value questions: condition on the first step and, when the remainder is identical to the original, solve the resulting equation.
E = 1/2(1) + 1/2(1 + E) gives E = 2 for the expected flips to the first head - no series summed.
Where it goes wrong. If your equations come out inconsistent, the state is underspecified. For "expected flips to see HH", knowing only the count is insufficient; you also need whether the last flip was a head.
Identifying the right state is the same skill as in Markov chains and dynamic programming.