We're going to start real basic here. If you don't already have a basic toolkit at home that includes a hammer, screwdriver, and pliers (including a needle nose pair), then you need to run to the store now and buy them anyways. Every household needs these guys whether you cosplay or not. As far as cosplay goes, these basic tools will help you bend, shape, and pry any number of materials.
For your cosplay toolbox you'll want a number of different paintbrushes. This includes size, quality, and type (foam vs. bristles). Often you'll want cheap, throw away brushes since they'll get ruined in the process (e.g. painting on a glue or latex based sealant). Other times you'll want higher quality brushes that will maintain their shape (e.g. detailed painting).
Safety equipment is often overlooked by new cosplayers since they're so caught up in the excitement of their new endeavors. Safety absolutely needs to be a forethought though! Just be smart about it -if there's any question in your mind about something, or others suggest a safety precaution, then by all means follow through yourself. You know what they say; 'better safe than sorry.'
Basic items include safety glasses/goggles/shield, particle masks or respirators (definitely invest in a quality respirator mask that fits properly), ear plugs, skin protectors (such as rubber gloves or long sleeves), and hair ties (for those with long hair).
Another type of equipment you'll want handy are various types of rulers and measuring mats. A self-healing cutting mat makes a great cutting surface, can help you square items up, and is great for when you're working with fabrics among other materials too. Measuring tapes and ribbons are great for larger items or ones that are 3D. You'll also want a metal ruler to help guide your knives when making cuts (if your cutting utensils are properly sharp, they will cut into/eat at a plastic rule; and we don't want that). A clear acrylic ruler is great for you when need more viability when cutting.
Talk to different cosplayers and you'll get different answers on which glues to use (both type and brand). Based on my research and participation in cosplay community forums, and through my own personal experience, these are the glues I would recommend having on hand. Of course, which glue you'll use will depend on your project, its materials, and various requirement such as time or money.
First I would recommend having a glue gun (metal tip preferred) on hand. This guy is pretty good all around as it can be used for gluing or detailing, and doesn't cost much. Having a basic wood glue or white school glue on hand is great for sealing items in preparation for painting. Mod podge (or Gesso) can also seal and smooth out surfaces for paint. E6000 is one of my go-to's when it comes to glue. It's an industrial strength adhesive that will give you a permanent yet semi-flexible hold. I also like to have my Loctite Super Glue and Fabri-Tac permanent adhesives on hand. These guys will give good hold and generally dry pretty quickly.
Cutting utensils and tools are a must for any cosplayer. You'll want a good pair of scissors for general cutting, a rotary cutter (often used with fabrics), an X-acto knife type blade for precision cutting, and a utility knife with break away blades (great for foam). I would also add a wire cutter to this list as well. Remember, always keep your blades sharp, or replace them once they become dull. You'll get much cleaner cuts this way, plus it's actually safer to use a sharp blade over a dull one.
A set of basic clay tools can also be very handy. These guys can help you shape, detail, carve, smooth, and more.
Nearly every cosplay you make will probably require a little bit of sewing in the least. A sewing machine will make your life a whole lot easier in this arena. I currently own the Brother CS6000i sewing machine and love it. It's super simple to use, so it's great for beginners, plus it has a number of excellent features, making it great for those who are more advanced too; plus you won't break the bank making this purchase. Of course you'll also want basic sewing accessories to go along with your machine such as various needles and presser feet (for different types of fabric material), pins, and extra bobbins.
The rotary tool is your all in one handy man. This guy will help you cut, drill, sand, detail, and more! Trust me, you'll definitely want one of these. I own the Dremel 4000-6/50 (which can basically do whatever I want it to), but you could just as easily start out with a scaled down model for less cost.
Another must have tool in your box is a heat gun. This tool will help you shape foam and mold thermoplastics among other things. I own the Wagner HT3500 and love the setting controls on it.
While other cosplayers may drop this from the list of basic tools (and opt for it as a secondary one), I would still recommend getting a hot knife too (although, if your budget isn't big enough for all the tools on this list, I would say to put this one at the bottom). A hot knife tool is another guy that can be extremely versatile. It can cut, puncture, smooth, and detail, as just a few of its uses. I currently own the Creative Versa-Tool.
Fellow cosplayers: Anything you would add or subtract to this list?
Future cosplayers or hobbyists: Any questions about any of the tools on the list? -More details about how or when they're used?
Please let me know in the comment below!