I recently made an attempt at video-making. If you'd like to see the video version of this very simple tutorial, click here.
(Edited April 23, 2016.)
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I have recently started to seriously use Google Drive for my job and was quite upset at first when I found out that making tables in Google Docs isn't as easy or straightforward as in Microsoft Word.
One of the problems I had was with splitting tables. Especially across pages. I finally found out how to do it.
All you have to do is to highlight the part of the table you want to split and drag it to an empty space above or below the original table. Drag and drop! By doing a simple drag and drop, you can finally split a table in Google Docs easily!
Step 1: Highlight all cells from the point you want to split. (I.e., if you want to split the bottom half of a table from the top half of a 2x4 table, highlight the bottom 4 cells.)
Step 2: Click and hold on the cells you have highlighted and drag to an empty space at the bottom of the table. (Use the "enter" key to create the empty space. A minimum space of 3 empty lines will result in an effective split.)
Step 3: Observe that your table has miraculously split!