In sklearn's classification report, is f1 the best accuracy measure? In the classification_report provided by sklearn, which score should I look at to make the best determination of the accuracy of my model? 

             precision    recall  f1-score   support

          0       0.70      0.68      0.69      5007
          1       0.65      0.54      0.59      2270
          2       0.37      0.22      0.28       614
          3       0.74      0.30      0.42       252
          4       0.59      0.42      0.49       262
          5       0.35      0.11      0.17       455
          6       0.34      0.23      0.27       248
          7       0.09      0.05      0.06       133
          8       0.38      0.15      0.21       395
          9       0.43      0.31      0.36       182
         10       0.23      0.12      0.16       230

avg / total       0.60      0.52      0.55     10048


As far as I understand it:


Precision tells us the amount of samples the classifier has correctly
marked as true positive out of all positive results.
Recall tells us about the number of samples the classifier was able
to get correct out of all samples in the set.
F1-score is the harmonic mean of precision and recall.


Maybe I'm misinterpreting the classification_report, but wouldn't the f1-score give the best view on the performance of the classifier?
You got recall incorrectly. It does not mean "correct out of all the samples".
$\hspace{50mm}$

Looking at the image,

$\hspace{50mm}$ Recall = $\frac{TP}{TP+FN}$

It is explained here.

From the above documentation, you can also see that you can modify your F-beta score to suite which to weigh more, precision or recall. Therefore, f-score would give you an overall of how good your classifier is, but if you need to know at which samples your classifier fails, you need precision and recall.