Here's the thing: a lot of art media can be dangerous if not handled well. Solvents, oils, and more, are very common in an art studio and all of them can be handled very safely, if you're being smart about it. If you're not being smart about it, though, it can be hazardous to you and/or those around you, including pets.
Some really basic rules:
1. Don't smoke and/or have lit candles. As a former smoker, and someone who happens to like candles, this is just something to be aware of. Oil, mineral spirits, etc. are very flammable, don't tempt fate.
2. Dispose of rags properly. Especially be aware of this if you paint with oil paints, as the improper disposal of rags that are soaked with paints and linseed oil can lead to spontaneous combustion. There are commerical products, such as the Justrite galvanized oil cans, but you can use metal or glass jars with water and submerse your rags/towels in them until you can safely dispose. Basically, this will prevent the oxidization process that actually leads to combustion under the right circumstances. Take it seriously.
3. Don't eat while painting. Even for other media, consider that the many pigments are harmful when ingested (such as the cadmiums) and while it is possible to get media that doesn't have toxic pigments, it just a really good safety habit to be in. What you do not want is to have your food cross-contaminated with your art media, ever.
4. Good ventilation is important, especially if you are using solvents. You need to have good air exchange to ensure any fumes are moving out in favour of fresh air. Use a fan to move the air, if necessary.
5. Consider using gloves or barrier creams. I've been a bit lax on this front, mostly because nitrile gloves are sweat inducers, but I've ordered some barrier cream for my hands. This will help protect the skin and, as an added bonus, make it easier to clean up. Do consider the gloves when first loading your palette because, if nothing else, it makes it easier to clean the cap threads to prevent sticking, but it will also let you start the session with clean hands.
Speaking of sticking... When it comes to paints, if the lid is stuck, don't just go reef on it with all your might, that's a mighty fine way to explode paint all over yourself and your studio. For oil or watercolour paints, you can put some very hot water in a jar and soak the top of the tube in the hot water for a few minutes to soften the paint.