What is the best method for classification of time series data? Should I use LSTM or a different method?

I would not use the word "best" but LSTM-RNN are very powerful when dealing with timeseries, simply because they can store information about previous values and exploit the time dependencies between the samples. That said, it is definitely worth going for it.

It has been proven that their performance can be boosted significantly if they are combined with a Convolutional Neural Network (CNN) that can learn the spatial structures in your data, which in this case is one-dimensional.

Take a look at this state-of-the-art method that combines LSTM and CNN, published in 2019.