Mealybug are whitish and have a white fluff around them. Scale is hard, semi spherical and brown to black. Scale and aphids both secret a sugar that ants harvest. You haven't seen a real scale infestation until you walk under an indoor tree and notice that your shoes stick to the carpet.
In the picture above the oldest adults are dark brown and the youngsters are light brown. They move out up and down the leaf vein where the flow of sap is the strongest.
Scale are difficult to eradicate as the hard shell protects them from soap and water. However a soft tooth brush dipped in soap and water would be effective in removing the adults. Use the toothbrush once and then wipe the leaves and stems, top and bottom, with a cloth or paper towel soaked in soap and water at five to seven day intervals to catch the eggs and stragglers.
High nitrogen levels will attract scale but the most likely source is that they came with the plant from the grower. If the grower only had them under control and not eradicated the scale will return as they can hide in the axils of the leaves.
Thanks to GradyPlayer for pointing out that the two are related. Even with their different lifestyles they are both hard to control on houseplants. As a former licensed pesticide applicator I found that if soap and water doesn't remove them then pyrethrums, neem oil or registered pesticides are not 100% guaranteed either. It's not what you use, it's the coverage. Can you get your choice of control agent to cover the entire surface of the plant and can you get the same coverage when you repeat the treatment at five to seven day intervals?
Edit: Magzalez asks
What are the white dots, and how can I get rid of them? They are scale insects and you can use a variety of methods but success involves covering the entire surface of the plant at five to seven day intervals with your choice of soap, neem oil or dormant oil. Best methods with scale involve physically removing them.
Are the ants all over the plant a bad thing, and, if so, how can I get rid of them without harming the plant? The ants are there for the plant sugars the scales excrete. Remove the scales and the ants will go away. They are farming the scale and are not directly harming the plant.