How do I get better at having tough conversations?

Congratulations on your new baby!!! That is such an amazing time for your family, and often the most trying.

I feel like I want to change the title of this question to "how do I get better at setting boundaries (specially with parents) ". It sounds like you and your wife have gotten to the point where you are feeling the need to set some different boundaries with your extended family and specifically your mom. It is a strange feeling as you grow older, and begin to realize that your parents are just normal adults too, and sometimes we mature past them and that feels even more complicated. Often times there is a level of guilt associated with setting boundaries with our parents specifically, they've provided us with so much. It feels like we can't be thankful for all they've done for us and also feel hurt by some of the things they've done as well. I know it's kind of a "hot take" you could say, it does sound harsh. If someone you love is crossing your boundaries and/or hurting you in other ways, you always have the right to set or reinforce those boundaries. Setting boundaries with friends or coworkers is hard enough, and setting them with family is another beast. Your priorities have changed to your wife and daughter, as it should be. You never have to explain yourself for that. So all of that to say- your boundaries are valid and you don't owe anyone an explanation for them. 

With that being said- the conversation still sucks. First, I would say depending on our relationship with our care givers, sometimes we know in what circumstances they are more likely to react positively-like when was the perfect time to ask to stay out past curfew for that concert. If you have that relationship with your Mom, start there. How do they communicate best-in writing? Over breakfast? Over the phone? Secondly, talk with your wife about doing it as a team, or talking to your mom just 1-1. There are a lot of reasons to talk about this, most importantly, your comfort levels. There is no rule book for these conversations unfortunately, and sometimes even when we prepare it goes off the rails and that's okay. Your boundaries are still just as valid!