There are some basic art supplies that every child should have. You probably already have a lot of them. Beyond that the possibilities are endless! Here is what you should start with:
Markers. I recommend the Crayola Ultra-clean washable type. These have ended up on my couch, curtains, floor and children. I can vouch that they come off easily.
Crayons. Good old crayola crayons work great.
Colored Pencils, and a pencil sharpener. If your kids are under 4 you may want to hold off on these, or put them away and only let them use them when you are supervising.
White paper. You could start with a ream of copy paper, and later upgrade to thicker papers for painting, and even special paper for watercolor. If it's in your budget buy a large roll of paper as well. I found some cheap ones at Ikea.
Construction paper in various colors.
Tracing paper. Kids love tracing paper and it helps them figure out how to draw forms.
Scissors. Depending on the age of your child and how much you trust them, you might purchase ultra safe plastic toddler scissors that only cut paper. A step up would be metal ones, just make sure they have a blunt tip.
Elmers white glue and a glue stick.
Scotch tape and masking tape. Preferably the blue painters masking tape, because it is easily removed. You will use the blue masking tape for taping down paper when painting.
Tempera Paint. You may want to start with crayola washable paint. When choosing paints, stick with tempera paint, because it's washable. Avoid acrylic and oil paint, these are not washable, and some have toxic materials in them. I buy student grade tempera paint from Blick in pint sized containers.
Paper plates for mixing colors on. Not the super cheap floppy ones, but the thicker ones.
A plastic jar full of doodads (pompoms, googly eyes, jems, buttons, etc). These will be a hazard for kids under 3, so I recommend keeping them out of their reach. I learned this the hard way after my 2 year old stuck a small bead up his nose. It's not just the little ones to worry about, make sure that you have a talk with your older kids about not putting things in their noses or ears. Since some of these materials could be a choking hazard you will want to avoid having them accessible to babies as well.
Watercolor Paints. There are basically two different types of watercolor paints, pan watercolors (for example the crayola ones that come in a plastic white pan) or liquid watercolors. I wrote a whole blog post about the two different types of watercolors, you can read more about pan versus liquid watercolors here. Q-tips and ice cube trays work great with liquid watercolors. Using waxed paper on the floor, then taping the paper down with blue painters tape makes for super easy clean up as well.
Sticker books and coloring books. These are not necessary, but you probably already have them and kids love them. It's all about getting kids excited about art! When my daughter was one we discovered the Melissa and Doug coloring books that come to life with just water on a brush, these are great for little ones because it teaches them how to hold a brush and how to control where they put it.
Paint brushes. Try to provide a variety of sizes, small ones are great, but also have on hand some larger house painting or foam brushes. Brushes come in all different price ranges, start with some cheap ones, when your budding artist turns into a little Picasso you can splurge on some higher quality brushes.
Plastic containers that can hold water when painting. You don't need to buy these, just wash out and save some old food containers, or find some old tupperwear that you don't use anymore. I like clear containers because kids love looking at the color the water changes to. Avoid glass jars, because they will inevitably drop it.
An easel if it is in your budget. Don't worry if you don't have the space or budget for this, kids are just as happy painting at a table or on the ground.
Sidewalk chalk.
Play dough in various colors. Buy it, or you can make your own and store it in tupperwear, just google it. You can probably find some tools in your kitchen drawers that your kids can use to play with the play dough (rolling pin, garlic press, meat hammer, etc.). Play dough could be a choking hazard for babies and young kids, so you may want to keep it out of reach if you have very young children in the house. Try not to freak out when they eat it, little ones love exploring the salty flavor, and a little won't hurt them.
Other things that your kids like. Let your kids have some say in the art supplies around your house. Sometimes my daughter will pick out something that I absolutely can't stand (fashion drawing light table), but it's not about me, it's about what sparks her interest. You could make picking out art supplies from Target or Michaels a treat for them!
Beyond purchasing all these art supplies, so many art materials can be found in nature and even in your kitchen cabinets. I'll write more about these and how to use them to create process based art in future blog posts!