Start with whatever rod you already fish with. Unless you're in a hut, you have room to use it.
The thing I love about ice fishing is you can be out in places you have to have a boat to get to when there's no ice. You can buy a handpowered auger for around $50, use a rod you use for summertime fishing, and buy a little ice scoop for a couple $'s. Throw a maggot on and you're fishing. You can make it cheap. Now the more money you spend the more comfortable and successfull you'll be. And the easier it gets to drill a hole. But you're fishing.
You're feet will be cold unless you invest in warm boots. But they don't have to be $400 mountain boots. A pair of Muck boots with some foot warmers in them should keep you warm.