They're not all a must and I'd say that some dogs don't need any of those tools at all. It is very dependent on the dog, what you plan to train the dog for (or how you plan to train), and what you are comfortable with. Either way, if you do decide to use these tools, make sure you find an experienced balanced trainer who can show you how to properly fit and use these tools, since it sounds like you're new to those collars.
I used the prong collar on my dog for a couple months, introducing it to him at 4.5 months, going through puppy class and obedience class with it, because I already have experience with the tool and Archer wasn't responding to other, less corrective methods. In my experience so far, he's a bit of a "harder" dog (although not drastically so) and sometimes needs the physical correction to get him back on focus or correct a behavior. (But FWIW I also had a very soft dog who never needed more than a verbal correction and she never needed anything more than a flat collar. Like I said, entirely dependent on the dog)
After 2.5 months of obedience class, puppy class, and my own training with him, I introduced the e-collar to him (when he was 7 months old) because his on-leash obedience was already pretty solid and I wanted to move him on to advanced obedience as well as eventually have him off-leash trained for hikes and such. If you want to have off-leash hikes or distance obedience with your dog, the e-collar is an excellent way to proof that obedience as well as use it as a tap or nudge on the shoulder if the dog isn't paying attention to you, which is how I use the e-collar with my dog.
I use Herm Sprenger prong collars with a loose backup chain collar, and I have the Educator 300 as my e-collar. Both are excellent brands, and I'd only recommend the Herm Sprenger collars if you decide to go with a prong. Educator and Dogtra are two highly-recommended brands if you decide to go with an e-collar, and whatever collar you use, if it's prong or e-collar, I'd highly recommend finding someone experienced with those tools to show you how to use them in conjunction with obedience training. A corrective collar is not a cure-all and if it's not used in combination with appropriate obedience training, it will be useless. Some people say that a dog will always be dependent on a corrective collar, but I haven't found that to be the case with appropriate and consistent obedience training.
And corrective collars, should you decide to use them, should not be skimped on and anything you find from Petsmart or Petco will not be the quality that you want to use. So prepare to spend $30-40 on a prong collar and upwards of $200 on an e-collar but trust me, you do not want to get the cheap prong collars or e-collars.
Again all this is my experience and I'm sure other people will chime in with their experiences of not needing to use any corrective collars at all, and if you already start to work with your puppy when she's younger, you will go far without needing a corrective collar yet. I got my dog at 4 months and he had some fear aggression issues as well as needing to build confidence, and after I introduced the corrective collars in combination with puppy class and giving him a lot of structure in his everyday life, I saw a huge improvement in his overall personality, attention to me, and confidence level. I've almost entirely graduated off of the prong collar and am now using a flat collar or slip lead with the e-collar as the sole corrective collar; we're repeating basic obedience now with just the e-collar and slip lead (to still give a little bit of on-leash pressure to help him continue to condition more to the e-collar) and then we'll progress to advanced off-leash obedience, again with the e-collar.
In the beginning when I was introducing the collar, he did have a prong collar and e-collar at the same time but after conditioning him to the e-collar, I have moved off of the prong collar, although it's still in my arsenal just in case I need more immediate control.
I don't have a need for a bark collar with my dog, although it shouldn't be automatically ruled out if the behavior warrants correction. Again, going through a balanced trainer should always be your first decision if you decide to go with a corrective collar.