How big should batch size and number of epochs be when fitting a model in Keras?

Since you have a pretty small dataset (~ 1000 samples), you would probably be safe using a batch size of 32, which is pretty standard. It won't make a huge difference for your problem unless you're training on hundreds of thousands or millions of observations.

To answer your questions on Batch Size and Epochs:

In general: Larger batch sizes result in faster progress in training, but don't always converge as fast. Smaller batch sizes train slower, but can converge faster. It's definitely problem dependent.

In general, the models improve with more epochs of training, to a point. They'll start to plateau in accuracy as they converge. Try something like 50 and plot number of epochs (x axis) vs. accuracy (y axis). You'll see where it levels out.

What is the type and/or shape of your data? Are these images, or just tabular data? This is an important detail.

Great answers above. Everyone gave good inputs.

Ideally, this is the sequence of the batch sizes that should be used:

{1, 2, 4, 8, 16} - slow { [32, 64],[ 128, 256] }- Good starters[32, 64] - CPU[128, 256] - GPU for more boost

I use Keras to perform non-linear regression on speech data. Each of my speech files gives me features that are 25000 rows in a text file, with each row containing 257 real valued numbers. I use a batch size of 100, epoch 50 to train Sequential model in Keras with 1 hidden layer. After 50 epochs of training, it converges quite well to a low val_loss.