Those are very good questions. Based on what I now know, if I were starting all over again, this is the collection of pizza equipment and accessories I would purchase for my particular pizza making needs:
Stand Mixer I would select a KitchenAid “Professional” (or equivalent) series stand mixer with a spiral hook (not a C-hook). My selection is predicated on the volume of dough I typically make, which is enough for one or two pizzas at a time. There are other mixers that may be better for much larger volumes of dough, such as the DLX or Santos, so you might investigate these models if you plan to make large amounts of dough. If you do a forum search on those mixers, you will find a lot of information on them. There are cheaper models of mixers than those mentioned above, but I would rather make the up-front investment in a good mixer rather than trying to save a dollar or two with a cheaper, less effective model. If I did not select a stand mixer, I would perhaps select a good 14-cup food processor. A bread machine can also be used, but it would not be at the top of my list. There are pluses and minuses with each type of machine.
Pizza stone. I would select the largest stone that my oven can handle with the door shut, while allowing a little space around the stone for air/heat circulation. Some members prefer using tiles, which are much cheaper than stones and can be used in many different configurations. I have used both and personally prefer a pizza stone, but if I were to recommend that to you over tiles, I am sure we will hear testimonials from the users of tiles. You might do your own review of the forum posts on this topic, and make up your own mind. Tiles are cheap enough that you can have both tiles and a stone at your disposal, as I have done, sometimes even in combination.
Peels. I personally recommend two peels, one wood and the other metal, with the wood peel being used mainly for loading pizzas into the oven and the metal peel being used to manipulate the pizzas while in the oven and to remove baked pizzas from the oven (the thin metal blade is easier to get under the pizzas). Some people use only one peel, wood or metal, for both purposes. That is far too challenging for me. I am willing to pay for an extra peel and reduce the risk of mishaps when handling the pizzas. Of course, the size of the peel used to load pizzas into the oven should be commensurate with the sizes of pizzas you intend to make. If you plan on making small Neapolitan style pizzas, you perhaps can get away with using only a small wood or metal peel. There is also a specially designed peel (called a SuperPeel) that can be used to load pizzas into the oven. For some, that may be a good choice, especially for those who have had difficulty in mastering the use of standard peels. However, I don’t believe the SuperPeel can be used to make large pizzas, such as 16” and greater.
Pizza Screens. Screens are very inexpensive and have a great deal of versatility and utility, especially in the summertime where there is little desire to crank up the oven to preheat a pizza stone (or tiles) for an hour or more. Screens also come in several different sizes. I have several screens of different sizes, although if I had to settle for just a single screen, I think it would be a 16” screen. With that screen, you can make pizzas of sizes up to 16”, which will work (i.e., fit) in most home ovens and is typical of the size of a NY style pizza. Screens also work well with the American style pizza. They are less often used for other pizza styles. Screens also avoid the necessity to use a peel since a pizza can be dressed directly on the screen and then placed in the oven. It’s so easy, even a caveman can do it.
Pans and disks. I like the idea of having a couple of pans on hand for certain style pizzas. My preference is for dark, anodized cutter pans, one unperforated and one perforated. If I could not afford a pizza stone, or simply didn’t want to use one, or to use screens, I would perhaps select a dark, anodized, unperforated cutter pan. With that one pan, I can make several different kinds of pizzas (e.g., flat pizzas, pan pizzas, Greek style pizzas, and even thin versions of deep-dish pizzas). Again, if I had to select only one size, it would perhaps be a 16” pan. My second choice would be 14”. Disks are also useful but, given a choice between a cutter pan and a disk, I would personally pick the cutter pan. Disks are primarily designed for impingement conveyor ovens, which have thermal and other characteristics that differ quite significantly from our standard home ovens. That said, I will soon be testing a special hearth-type perforated disk (16”) that was designed to be used (in commercial conveyor ovens) to simulate a NY style pizza that is most commonly baked in a deck oven. I will be using the disk in my home oven.
Digital scale. For me, a good digital scale is at the top of my list of equipment to be used in the course of my pizza making. I tend to be numbers oriented, and I find that using a digital scale gives me consistency, uniformity and reproducibility in my pizzas. I also do a lot of experimenting with dough formulations, where a good scale is a necessity for me. However, I am also mindful that others prefer to use volume measurements and “touch and feel” in making their pizza doughs. Most members tend to align themselves on one side or the other on this issue, and you will occasionally see spirited debates on the forum on this subject. I don’t have a quarrel with either position. We are all different.
Other/Miscellaneous items. I would also recommend that beginning pizza makers have a collection of other pizza-related items such as a sturdy pizza cutter, a good set of measuring cups and spoons for measuring dry ingredients, Pyrex-type cups for measuring liquids, a good set of mixing bowls, dough storage containers, an instant read thermometer, a bench knife, and a few pot holders. I would also have a few spatulas, including a thin, long-bladed flexible plastic spatula that comes in handy when using a stand mixer to combine and mix ingredients. You will perhaps also want to have a wire rack or two on which to place your finished pizzas, and a flat pan or two on which to cut and serve your pizzas. If you were interested at some point in making deep-dish pizzas, I would recommend a dark-anodized or seasoned deep-dish pan, and a pan-gripper. If you elect to make cracker-style or thin and crispy pizzas, I would recommend a dough docker.
No doubt I have missed an item here or there but the above reflects my preferences off the top of my head. Others are bound to have their own recommendations, which you of course should consider before rounding up your own personal collection of equipment and accessories. Of course, there are likely to be budgetary considerations that you will have to take into account in making your selections.