Congratulations!! While I know you may feel overwhelmed right now, we are very excited for you!
Like TexasRose, I wonder if you have family in town to help out. That would certainly be a huge benefit. However, if you don't, don't fret, it can still work out. You might want to look into a combination of day care and part-time nanny. Day care for the day (they usually will take kids as young as 6 weeks), then you'll need a nanny for the nights on-call, weekend when you work, and possibly to help with day care drop-off and/or pick up, depending on what your hours are. Or you could go with just a nanny for the time that your fiance is gone, then do day care when he gets back and can hopefully cover the nights and weekends.
I won't lie - any arrangement is going to be expensive. My daughter's full-time day care is about $7k a year. Adding a nanny can be anywhere from $10-20k. But you'll both be making money and working so much that you won't have time to spend it (trust me).
Good luck, and let us know if you have any more questions!! There are lots of us out here who have done med school and residency with kids!