Good techniques for stirring dough
You generally don't stir cake batter at all.

Where I learned to bake, we were taught that if you want a light and fluffy cake, as is generally the case, then you should fold in the dry ingredients (including flour).  That means using a flat surface - a silicone spatula works great but you could even use your hand in a pinch - sliding it down the edge of the bowl, and using a turning motion (i.e. folding) to incorporate the flour, repeating several times until there are no longer any large clumps of dry ingredients.

When you fold, it's very difficult to over-mix.  And particularly with cakes and quick breads, under-mixing a little is actually OK, because the batter tends to be quite moist and eventually the moisture will seep through to any unincorporated flour - and if it doesn't, you'll get a nice spongy texture.

Commercial mixers actually have paddle attachments for folding large amounts of batter.  You don't need a special mixer, though; you just need to be gentle and conservative with your mixing.

Not every cake is the same, and some recipes may specifically call for you to whisk the batter (for which you should use a balloon whisk, not a spoon or spatula) instead of folding, but if it just says to "mix" the batter then I would using the folding technique.