Some mares are harder to milk than others and as stated, not all mares bag up prior to foaling, but most do. There is never going to be one solution that fits all mares. I personally prefer the mother natures strips as I can always get enough milk for the test and its so darn reliable! That reliability is totally worth it to me.
Never milked a cow, but my guess is its similar. Just start up above the nipple, in the bag area, apply even and firm pressure as you pull downward. It can take a few strokes before the milk comes out. Practice getting milk out successfully before trying to test it. Just get your technique down and then try capturing some on a shot glass. This method works for me.
Good luck!