How do I determine what max 3 way bulb will be safe in my lamp?
If it says 60 watts max, then the maximum wattage bulb you can safely use is 60 watts. If you could find a 3-way incandescent bulb like a 20-40-60, that would be safe. But I've never seen one that small.

With such a low acceptable wattage, the lamp may have been designed with CFL or LED lamps in mind, you can get something like an 11-23-34 Watt CFL 3-way bulb that gives light equivalent to a 50-100-150W incandescent. That would be perfectly safe in this lamp,  assuming that it fits inside the "harp" or whatever holds the shade on.

3-way LED's are a little harder to find, but they do exist and they often are closer to a standard light bulb in size and shape, so they may fit better in your lamp.