This is one of the cases where every answer should start with:
Every child is different.
There will always be those that sleep though the night almost from day one and others that need more time. A lot more. Some will need the touch or smell of their parents, others are happy to simply fall asleep. And as different as the babies are, as manifold are the "surefire recipes" to avoid the disturbed nights and lack of sleep that are so characteristic for early parenthood.
But to answer your question:
As far as any formal sleep training goes, three months is way too soon (and I'm not a fan of it, but that's my personal opinion).
There are two things you can and should work on, though:
- teaching your child the difference between night and day and
- falling asleep again as fast as possible.
For your child, that means during the night, there will be only a minimum of activity from your side. Keep the lights low, voices down and change the diapers only if necessary. Make the difference between day and night as clear as possible. Playtime and activity belong to the day. Your child will learn pretty fast that while you are there to fulfil their needs (security and food), there will be no entertainment, so they might as well fall asleep again. They should be tired anyway.
For you as parents, this means you will be able to go through the whole "wake up - feed - put baby back to sleep" routine while being half asleep. As human beings, we can easily handle short interruptions of our sleep (bathroom visit, anyone?) and still be more or less well-rested in the morning.
Breastfeeding has the huge advantage that you don't have to get up and prepare a bottle in the middle of the night: after that, both you and your baby (that hated having to wait) are probably really awake, which can make falling asleep again harder and thus decrease the perceived quality of your sleep. For us, this also meant co-sleeping was the way to go, especially with the child that needed us physically nearby to sleep well. During the nights, I was able to nurse without having to get up and I'm sure there were cases where I fell asleep even before the baby.
Also please remember that many sources recommend "feeding on demand" (as opposed to a fixed schedule) to ensure a milk supply matching your baby's needs and yes, that includes night feedings.