It is not uncommon that residents need some sort of medical treatment during their training. Often, residents need to decide whether to be treated at their home institution, which affords greater convenience at the cost of some loss of privacy. The choice is ultimately yours, limited by what your medical insurance will cover.
I can't speak to how time consuming the whole process is. I expect one challenge is that your treatments will need to be timed with your cycle, which will not allow for much flexibility. Still, this is something which hopefully can be worked around (especially if you're at your home institution). I also expect the time commitment will depend on whether you're successful with an early cycle, or a later cycle.
As to how to tell people, that's up to you. It's sad that you think you need to do "damage control" here. Hopefully, much of this can be handled while maintaining your schedule. Obviously, when successful, you'll need a maternity leave but that's not uncommon. Most women undergoing fertility treatments don't want to talk about it with the world, as it often leads to "how's it going" which, if things are difficult, is a tough subject. I would hope that residents at my own program would feel comfortable telling me about it, simply so that I understand when they need schedule flexibility.
