I'm impressed with all the answers here. This is not a new answer, just an attempt to summarize the timings of all these methods. I considered the case of a series with 25 elements and assumed the general case where the index could contain any values and you want the index value corresponding to the search value which is towards the end of the series.
Here are the speed tests on a 2013 MacBook Pro in Python 3.7 with Pandas version 0.25.3.
In [1]: import pandas as pd
In [2]: import numpy as np
In [3]: data = [406400, 203200, 101600, 76100, 50800, 25400, 19050, 12700,
...: 9500, 6700, 4750, 3350, 2360, 1700, 1180, 850,
...: 600, 425, 300, 212, 150, 106, 75, 53,
...: 38]
In [4]: myseries = pd.Series(data, index=range(1,26))
In [5]: myseries[21]
Out[5]: 150
In [7]: %timeit myseries[myseries == 150].index[0]
416 µs ± 5.05 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)
In [8]: %timeit myseries[myseries == 150].first_valid_index()
585 µs ± 32.5 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)
In [9]: %timeit myseries.where(myseries == 150).first_valid_index()
652 µs ± 23.3 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)
In [10]: %timeit myseries.index[np.where(myseries == 150)[0][0]]
195 µs ± 1.18 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)
In [11]: %timeit pd.Series(myseries.index, index=myseries)[150]
178 µs ± 9.35 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)
In [12]: %timeit myseries.index[pd.Index(myseries).get_loc(150)]
77.4 µs ± 1.41 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)
In [13]: %timeit myseries.index[list(myseries).index(150)]
12.7 µs ± 42.5 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)
In [14]: %timeit myseries.index[myseries.tolist().index(150)]
9.46 µs ± 19.2 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)
@Jeff's answer seems to be the fastest - although it doesn't handle duplicates.
Correction: Sorry, I missed one, @Alex Spangher's solution using the list index method is by far the fastest.
Update: Added @EliadL's answer.
Hope this helps.
Amazing that such a simple operation requires such convoluted solutions and many are so slow. Over half a millisecond in some cases to find a value in a series of 25.