I have put 38c tires on a 20c rim but it was kind of hard to get the bead to seat. Once it did it was fine. I think you could get 32c tires on the 18c rims. Maybe even some 35c tires. Some studded 32c tires would be just fine. No need to go any wider unless you plan on riding without studs.

I actually prefer 32c tires since they are lighter. 37 &38c tires work better with a 23-25mm wide rim which are generally the width of entry level 700c wheels and touring wheels. 35c tires are kind of in between size but I think you might have a little trouble with a 18mm rim.