You need to ask the rink manager/skating director for permission to use a marker, even if you see coaches using them. Some rinks prescribe the brands/types of markers, others don't allow them at all.
I use Crayola Washable markers for toddler classes - lots of colors, doesn't bleed into the ice, washes out of their clothing easily. For groups, I use Marks-a-Lot Washable markers, which are hard to find, but work well. Don't use Rose Art brand - it bleeds deeply into the ice.
In a pinch, I use Sharpie Magnum or Avery Wide Tip permanent markers, which last longer but ... the tips wear out faster and the slush will stain clothing and skates. (I have lovely blue spots all over my white skates.)
Always test the markers first on a hockey line - just draw a squiggle or line, let it dry, then scrape it off with your blade edge. If it doesn't come off easily, the zamboni won't be able to clean it off, so don't use that marker.
Tap the marker after using it, to shake off the ice and stop the drips. Make sure the cap is on tightly before you stow it in a pocket. Fabric will draw out the ink in the marker, leaving you with a messy jacket and a useless marker.