How can I check if a bigger training data set would improve my accuracy of my scikit classifier?
One idea:


Split your data into train / hold out datasets.
Train the model on a fraction of the training data (say 50%) and test on the holdout dataset.
Train the model on a larger fraction of the training data (say 75%) and test on the holdout dataset.


It's important that you use the same holdout data for testing so you can perform a true test of accuracy.

Since you're doing classification, you should check that your data is balanced, and adjust if not (this may also improve your accuracy without needing larger training data).