I have long been a shooter of the double action trigger, the first pistol I fired at age 10 was a double action .22. I still remember the horrible double action pull on the Erma .22 pistol it was. None the less I grew to love and enjoy double action triggers, in either semi-autos or revolvers. These days, I carry and focus primarily on double action revolvers, with a bit of DA/SA semi-auto pistol shooting thrown in. I've shot all types of actions now, some good, some bad, some somewhere in between and I want to focus on some of the pros and cons of trigger types in your gun.
Let's start with the inbetweeners, these are your safe action Glocks, your striker fired XDs and M&Ps, basically most of your polymer framed striker fired guns. The triggers on these guns are often a bit of a mix, long and a little mushy, requiring a press that weighs somewhere between a single action and double action trigger pull (about 5-7 pounds of pressure). For the most part the triggers on these guns are serviceable, often benefit from contact point polishing, and generally can be smoothed with shooting. They best thing about striker fired guns is the trigger pulls are consistent from first to last shot. They take practice to master, but don't require you to be an IPSC master to have good control. My limited experience with striker guns tells me a few things, first none of the ones I've fired stock where "great" triggers. They often lack great feedback on the press and on the reset. Which leads me to my biggest complaint, sometimes the lack of feedback on the rest makes the gun difficult to run. My father who has arthritis in both hands and especially in his right index finger, cannot feel the positive reset on a stock Glock or M&P trigger. It is all but impossible for him to effectively "run the gun". With a Glock he must take his finger completely off the trigger to attain a positive reset, something that is not a positive for shooters. Not everyone suffers this of course, I have no trouble detecting the reset on a striker gun while shooting, but it doesn't give me the confidence that a crisp reset gives me. Another complaint from me is that because nearly all striker guns must have the striker "re-cocked" during dry practice, you have to adjust your technique and continue to hold the trigger down after squeezing, while you run the slide, and then release the trigger to feel the reset, it's a technique adaptation. You must do something similar on single action or traditional double/single action guns. The benefit of those guns though is there is usually an exposed hammer that is easily cocked with the off hand, as opposed to the slide movement required to reset a striker.
Now onto single action triggers, commonly found on your SAO (and traditional DA/SA) semi-autos and for those who are packing single action revolvers. The single action trigger is often touted as the greatest trigger known to man, I suppose that can be true. Good single action triggers are often good to great, not so good triggers are terrible. Because they often have little to no slack, a heavy single action pull can result in an unwanted jerk or an overly heavy press that moves the muzzle end of the gun unexpectedly, often during the moment of firing. It has been my experience though that many single action trigger pulls are light, some are mushy, but many of them are crisp. Generally the light, crisp, straight trigger pulls on 1911s and Browning Hi-Powers result in good shooter confidence, particularly in new shooters. They are easy triggers to master, being so simple. They also do not seem to be as affected by bad trigger control (particularly slapping), as a double action or striker gun would be. Greater trigger control is generally found in a single action trigger, because of its attributes, they also are the pistol trigger that most readily resembles the triggers on long guns of all types. My general complaints with single action triggers are that the mushy ones suffer the same problems as striker guns, they don't offer positive feedback during the press or on the reset. The benefit to single action triggers are that the resets are often very positive and tactile. My father mentioned above, does not handle a mushy trigger well, but conversely he handles a 1911 just fine. I've also found that if one is not going to practice a lot with their weapon (either dry or live fire), the single action trigger is the easiest for folks to pick back up and shoot well. It's straight forward, simple, and provides great and easy to understand feedback to the shooter.
Finally, the double action. As mentioned above the double action is my preferred trigger of choice and you may think I saved the best for last, not necessarily. The double action trigger has pros and cons as well. The primary pro of the DA trigger is that it is a very positive trigger. Because it has a long stroke action to it, it feels very positive as you start and finish your stroke. The flipside is, depending on the gun sometimes a double action revolver pull can change during the stroke. Colt revolvers are a classic example of this they exhibit a lot of "stacking", where the trigger starts off as "light" (7-10 pounds), and stacks sometimes noticeably sometimes not, to a heavier weight, before suddenly and abruptly breaking, firing the gun. A long, heavy pull, that stacks, is a very difficult trigger to master. For a the record Smith and Wessons stack slightly, but not a lot, Rugers have almost no noticeable stacking. Of course the benefit of the revolver trigger is again that it is positive and this is true on the reset as well, most revolvers have a very positive reset that is easily interpreted and understood by even novice shooters. The cons are the additional weight of double action pulls and the length, both of which can present problems for shooters with lower hand strength and shooter fingers. Most double action triggers are very positive though, I've only met a few in semi-autos that were mushy of any kind and even there, the weight was generally consistent to give you a more consistent firing stroke on the first shot. My experience has shown me that double action triggers are not the appropriate place to begin (as I learned at that tender young age), the heavy pulls often cause jerking or over exertion on the part of the shooter onto the trigger, tending towards poor accuracy and low shooter confidence. On the flipside, though DA triggers are harder to master, I have found that once mastered, the shooters tend to possess a higher degree of trigger control and manipulation techniques than his counter part with the same amount of time behind a striker or single action trigger. Once you have mastered the double action pull, you generally find that the single actions with the crisp, light, straight pulls are very easy and striker guns which fall somewhere in between are easier as well. With a double action pistol you must learn to truly control the trigger, because all mistakes you make are amplified to a higher degree than with other trigger types.
Overall, I encourage you to own guns of ALL the common types. I'm a guilty offender in not owning a striker fired "in betweener" trigger, but I am working on rectifying that situation. For the most part, I feel that the single action trigger, being the most consistent and generally easiest to use is probably the best of the bunch, but spending your time working other trigger types can be very beneficial. I recommend highly that you invest time in building double action shooting skills, particularly with double action revolvers. The skills you get learning to run a DA revolver will transfer almost immediately into better shooting with other pistols. You will find your lighter, straight pulls to be easier, less distracting to work on and focus more intently on getting your front sight onto the target and getting the press right. Also the DA trigger helps with follow through, because your follow through is the reset on a DA revolver and it helps you maintain a more consistent follow through, which also translates to other pistols.
I also want to add that most pistol shooters are often excellent rifle and shotgun shooters as well. Once you have mastered a heavy pull on a pistol, the lightweight, consistent, and constant pulls on rifles and shotguns will be that much easier. That's also really the case for the DA revolver trigger, once you've mastered the long, heavy, stroke, so that it is a consistent repeatable trigger press, you will find the shorter, lighter pulls much easier as well.
Good luck, be safe, and shoot a bunch!
-Rob