You need to create the initial surface coat that can both smooth the rough surface of the Styrofoam and create a surface for further material adhesion. There are actually a number of safe and simple approaches that you can do which all have a similar outcome. First though you need to be aware of what you should not do.
Be wary of products that will dissolve styrofoam
There are a number of products that react to Styrofoam on a chemical level and cause it to melt. Many spray paints can do this as well and products that contain paint thinner. Many product labels will mention what materials they are and are not designed for. If you are not sure sticking with one of the suggestion below would be advised or at least test in a controlled and well ventilated environment if you must.
Initial coating options
It does not need to be any sort of heavy coating but something that will try and provide a more uniform surface. In no particular order:
Mod Podge / decoupage: Mod podge will work on its own with one or two layers built up. Some light sanding in between layers will help create a smoother surface. You don't need to use the glue alone. You can decoupage as well.
Paper mache: Using the foam as a base paper mache onto it. I would likely do the entire object (all sides) as this might not adhere to the foam well or at least not long term. Doing all sides would give it no where to go. Not my first choice but this is a fun option for kids.
Styrofoam safe paints: I don't have any specific suggestion here. You will have a hard time using a brush given the rough surface. You might need to use a cloth or something similar to work the paint into all the holes. Let it dry and repeat the process. It will get easier each time as it gets smoother. Water and Acrylic paints apparently work just fine here.
Plaster of Paris / Geeso / Drywall compound: These should be easy to apply to the surface. Don't put it on too think. Much like other approaches Let dry and sand. Drywall compound would be the last choice of these options as it would be brittle. Should still work though. I am putting Geeso in this list as I think it works almost the same although I have no experience with it.
Commercial Options: There is a specific product called Foam Finish which is marketed for use with Styrofoam and related foam products. Likely there are similar products under different names. Consulting a hobby store will help you get the variants. From the container you can see it would work well in this situation and you can read more at one supplier's site. :
- Repairs dents & nicks in all foam including Depron EPP, EPO, Styrofoam & more.
- Helps make foam stronger with virtually no extra weight.
- Creates a smooth surface for decals or paint.
In almost all cases the key is to keep your initial coats thin and build up to strength and uniformity you desire.
All of these should get you a good working surface that you can paint whatever you want. You should be able to glue object as well to the new surface of your foam.