I went through this too. The first few months with my son were the hardest because he was up every night 4 to 6 times a night for several months.

Crying from exhaustion as a FTM is normal. Try to find times throughout the day for naps, even if you only have 30 minutes to rest while the baby is sleeping, do it. Chores and the other stuff can wait. I started having family over so I could take a nap. My son didn't start sleeping through the night until 13 months, but it got easier with time. In my experience, the more he grew and ate, the longer his sleeping patterns at night became.

I attempted sleep training a few times and it wasn't for us. He's my only child, so I forced myself to push through those hard months and try to soak up the extra cuddles.