Do your pins have a groove on one side? Better ones do. If you don't, look for some. The pin inserts with the string inside the groove. I push the pin in making sure the string is in the groove, then hold it down with my thumb and give a little tug on the string til I feel the ball 'lock' the pin in place, then I go about the business of stringing. Note that the wound 3rd-6th strings are always tougher to work with.
Another issue is that not all bridge pins are of the same diameter; if too thin, then they'll never fit right, always slip up. If too thick, they won't go in deep enough. It's possible from the string/pin slipping or popping up/out as you describe that you may have damaged/chipped the bridge plate (slat of hardwood on the underside surface of the top opposite the bridge. This can be fixed by a competent luthier if not too mangled, otherwise, you'll need the plate replaced and that repair will suck your wallet.
Bridge pin pullers are surely the best tool, NOT pliers. I have a different one than that Dunlop and never use the POS fattie on a string winder. But I always place a polishing cloth over the bridge before sliding the puller under the head of the pin to avoid denting a rosewood bridge or marring an ebony one.
Finally, the pin material is a factor, and it can also improve or alter tone. Tusq or bone are better and far more durable than common plastic pins and are usually stock on better acoustics, just as are nuts and saddles of the same material. they hold better because they don't wear out so fast (the outer, cotact surface and ball tip end. Plastic pins tend to bend over time because they're soft, brittle. They also put a bit more meat in your treble end frequencies. Wood pins are too soft. You can get brass if you want to add treble, and they hold up, but IMO, the sound is just too shrill.
Like anything else, you can go cheap with plastic pins for a few bucks and save now, pay later, or spend $10-20 on very good pins and pay now, save later. The more you play an acoustic, the wiser the investment. If it's only a now & then axe, probably not worth the expense.
And...as long as we had a topic a few days ago about Ovations, there's another pro to those: pinless bridges. Takamine makes some, too. The strings make direct contact with the bridge and the plate, yielding max top vibration transference and optimum sound.
^^
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