Surprisingly, it doesn't cost very much to set up your own little painting studio. You can get everything you need for under $200 but if you prefer more premium supplies and equipment then it will cost more of course. If your goal is oil painting, for example, then the one thing you don't want to skimp on is the quality of your brushes.
For either oil or acrylic painting, you will want to get a reasonably good easel. Although you will be able to find plenty of affordable "A" type easels, "H" frame easels tend to be more stable and sturdy. Typical starter easels will range anywhere from $80-600.
Good lighting. This can be either a bright space if you paint during the day or a good set of interior lights (preferably daylight-balanced) if you work at night.
Unless you don't care about the longevity of your paintings, try to get medium or "artist" grade paint rather than student grade. Buy large tubes for: Titanium White and Yellow. Also buy medium tubes of the primary and secondary colors. You can mix any colors you need from those as a base. Some colors will have different consistencies from others so depending on your painting need or style you may need to get additional variations of some of those bases. For example, in oil paints cadmium red will be more opage and alizarin crimson will be more translucent. They both have a strong red base (cadmium being brighter and more pure) but because of their different consitencies you will end up using them in different ways while painting.
To hold your supplies while you work. Any flat & stable surface will work.
Most canvases will suit both acrylic and oil paint. Most canvases you can buy from an art store will also be pre-stretched and primed, ready to paint. Be aware, however, that pre-stretched canvas is often not going to be well-suited for future re-stretching and may be of lower quality not suited for archivability. In the case where you care a lot about that, you will want to buy your own unstretched canvas (unprimed with primed gesso) and stretch it yourself onto stretcher bars. This isn't the easiert thing to do, it takes patience and experience to do right, but you gain the most flexibility with your work doing things that way.
Gesso is very important for priming your canvas, whether you intend to paint with oil or acrylic. You can skip gesso if you always buy your canvases pre-primed.
Both oil and acrylic have a lot of mediums to choose from. Different mediums have different uses, some are for extending the paint or improving the paint flow, some are for assisting in creating a base for mxing, some are intended to assist in thinning or glazing, others for texture control (referred to as "impasto") and finally others might be to help in controlling the drying time (whether speeding up or slowing down). At the very minimin you will want a thinner, a mixing medium for glazing, an impasto and finally a varnish as your final coat.
There's a common misconception that painting requires sable or natural-bristle brushes, and they tend to be the most expensive. However, after years of experience I've found the best type of brushes tend to be white synthetic which are interchangeable between oil & acrylic. They are bouncy, last a long time (if you keep them clean), and hold paint really well. They also tend not to lose bristles and easier to clean. They also keep their shape better. The most basic types of brushes to get will depend on your painting style but you'll want to have at least 1 of each of the following: a large/wide brush with stiff bristle, a medium flat brush, a very small flat brush, a small & medium round brush, a small and large fan brush, a small and medium "filbert" brush (it's flat but rounded on the end). As you progress with your painting, you will begin to learn what works best for you.
Usually used for mixing paints, can also be used to paint with if that's your style.
For oil paint this will usually be turpentine, for acrylic it will be just water (keep the brushes dipped in water while you're not using them if they're loaded with paint). After you finish your painting session, you'll want to thoroughly clean your brushes with the primary cleaner and then use a soap-based brush cleaner immediately after. Rinse with water and then re-shape the brushes with a way brush soap so that they dry in the proper shape. Before using those brushes again, just dip them in your preferred cleaner and wipe dry with a paper towel or cloth.
This would be a surface that you mix your paints on before brushing them onto your canvas. To keep your pallete re-usable without significant cleaning in-between, clip wax paper onto it first and squeeze out your paint onto the wax paper.
You'll want a large container to hold all of your smaller supplies, like paint tubes and brushes, and small/round containers for holding cleaners and mediums for use while you paint. The best large containers look like a tackle box used for fishing. Containers intended for toxic liquids such as turpentine should be glass-based and be carried with care to avoid dropping. Containers for holdign your brushes upright are also useful, giving you easy access to brushes whether they're in use or still clean.
An artist apron is always a good idea, or wear clothes you don't mind accidentally getting paint onto.
Paper towels, rags and regular towels should always be available before you start painting. Brush soap and turpentine are also important.
Pencils, sharpener, erasers