Buy a pair of Black Diamond Venoms. 50 cm hammer, 57cm adze. Not too much over your $100 price point.
They will get you up most anything you want to climb, from easy snow slogs to technical alpine routes, to waterfall ice. By the time you are tackling routes that are beyond the capabilities of the Venoms, you won't have to ask for advice about what your next set of ice tools should be.