Both paint and stain are safe to use, it is the type that you should be careful of. Always use water-based products as they as non-toxic. Another term that is used to describe water based products is latex. Stay away from anything that is described as lacquer, alkyd, oil, alcohol based as these products can be very harmful to your cham. They off-gas even after they've dried and are not safe to use in a living environment. Also, most spray paints are oil or alcohol based therefore should not be considered.
Its ideal to use an exterior grade paint, or stain/top-coat. As these will perform ideally in a humid and warm environment.
Note that these products take about a month to fully cure but can be used after 2 weeks. This will allow the product to adhere to the surface properly before exposing it to elements such as water, heat, moisture, etc.