If you fall through lake ice while your speed is low or zero, you probably can find the hole in the ice right away; so the first danger, of drowning, should not be a problem.
Whenever you go onto ice, you should have ice claws hanging around your neck, so you can get out again:
Photo credit: Holger Ellgaard, Wikimedia Commons, CC-BY-SA 3.0
You should be able to buy ice claws in outdoor or fishing supplies stores in regions where ice fishing is a popular activity.
If you are in a remote place and the shore does not immediately provide access to a warm and dry place, you will additionally need:
- A dry set of clothes to change into,
- means to make a fire.
Someone I know fell through the ice while ice skating in Vistasvágge in northern Sweden. He was alone and was quite far from the nearest road or habitation (I think some hours by ice skates). He climbed out using his ice claws, went to the shore, made a fire, stripped naked, dried himself, put on his set of dry clothes, and then returned to civilisation (I don't know if he skated back or walked back). He didn't see any other people while out skating that day.
Ice fishing should be safer. Probably, in many cases, people ice fishing are close to a home or a car when they get back to the shore. In this case, there may be no need to make a fire.