I'm writing this as a first time mom at a ridiculously early hour. Adjusting to being a parent has been so great but definitely can be draining especially from the lack of sleep. If I could give advice from my experience to new moms, here are some tips to get some sleep with newborns.
Sleep when baby sleeps. Everyone has given me this advice and they are right. Now having a newborn I can't plan when to nap or sleep (or do a lot of things) so napping at least once a day when he sleeps helps give me the extra energy especially at night when he is up. Many mothers will try to get things done when their baby sleeps (including me) during the day which can easily add up to a drained and exhausted mommy. So again, if you can squeeze in a once a day nap you will feel so much better in helping your little one.
Bonding time with baby in my arms or on my stomach gives me time to nap or at least relax. Even if he doesn't sleep and is looking around, I can engage with talking to him or singing which many times puts both of us to sleep but if not I can get a little relaxing time while bonding with baby.
A newborn doesn't know day from the night. Slowly developing patterns to show him day and night (for instance at night be less engaging, keeping lights lower, etc.) can help slowly teach him when bedtime should be. My husband and I have started to do this when our baby was about two weeks old. He definitely didn't sleep through the night, but he's sleeping longer strides at night and less during the day so hopefully it will continue like that.
A few hours before going to bed, we try to keep our baby awake so by the time he eats next, he is worn out and really wants to sleep.
Find out the things your baby enjoys to fall asleep. Our baby loves soothing music to fall asleep. Some like to be rocked, sang to, etc. I think our baby is starting to realize that after being fed and burped then turning on his little soundtrack, it's bedtime routine.