I have this Ikea coat hook, that I'd like to mount to a door. But I would like it to be rather sturdy to handle some bigger winter coats.
What kind of screws are recommended for (what I think is a typical) internal door?
I've used hollow door anchors for installing robe hooks on the back of two bathroom doors. They've been up for a year now with daily use, and they're both holding strong. The heaviest load they've seen is only a bathrobe and a towel at at time though, so I'm not sure they'd stand up to holding a heavy coat rack and 4 winter coats hung on it.
Home Depot sells these in packs of 4, just make sure you get the ones specifically made for hollow core doors. The stud solver drywall anchors with the same design are too long to fit in a door without going through both sides.
As Steven points out in his answer, these make a relatively large (~1/2" diameter) hole in the door as well. Unlike toggle bolts, you don't have to worry about the wings having room to open.
If your door is hollow (easiest way to find out is to knock on it) then you will need to use toggle bolts.
The downside to these is that you need to make a relatively large hole to get the bolt through. You might be able to get away with smaller or different anchors, but you specifically mentioned that you wanted to hang heavy objects, and these are best for heavy loads. Make sure that the screw is not too long otherwise you'll never get it tight enough as it will hit the other side of the door.
If the door is solid then you can just attach it with wood screws (#6 or #8, 1/2 or 3/4" will do the trick). You should pre-drill a small hole prior to prevent any splitting/cracking in the wood.
In both cases you will end up with holes in your door which is often undesirable. They also make hooks specifically for doors that hang from the top of the door instead of requiring anchoring into the door itself.
The other thing I'd mention is that inside doors aren't usually designed to hold loads like this. You might find that overtime the door will shift. You can probably mitigate this by driving 3" screws through the hinges on the wall side so that they are held up by the stud and not just the door frame.
There are Interset which are typically used for just this type of job (assuming your door is hollow & made from thin ply or hardboard etc).
One thing to note is that the load is spread in four directions thus making them more suitable for heavier loads on thin materials than the two direction or screw in types (less likely to rip through the material).