This thing was coated in years of dust and grime and spiders, so it took me a good twenty minutes just to give it a sponge bath. So, this is what it looked like after I cleaned it...still not good.
Once we finished sanding down the wood filler, we lightly scuffed up the entire surface area with some super fine grit sandpaper and then wiped down all the surfaces again, making sure they were free from dust.
We primed the dresser. Primer makes the paint adhere without having to sand down the wood like a crazy person. Look for a primer that says 'adhesion' on it, like the one I used here.
We used an angled paintbrush and a mini foam roller, brushing the paint on with the paintbrush and smoothing it out with the roller if needed.