I have a 5-month-old and a 2.5-year-old so I'm re-living a lot of this right now! My older son napped on me a lot, but obviously I've been forced to do things differently this time. My baby sleeps much better (though still not wonderfully, ha) because of it.
The easiest "routine" for me is remembering that until about 6 months, 2 hours is about the max awake time. So whenever my son wakes up, I note the time. After about an hour, but definitely before 2 hours, I put him down at the first sign of tiredness (yawning, cranky, rubbing eyes, etc). So put the white noise on, swaddle, feed, burp, and put in the crib or swing. The first few times I did NOT believe he'd fall asleep... and then he did and it was so magical.
My baby loves to watch my older son play and it's so much easier, I feel like when it's just you and the baby you feel like you have to be their whole source of entertainment! But remember that 5 weeks ago he was floating around in darkness 100% of the time so it doesn't take much to entertain ;) Put him where he can see you do your thing. Go outside or run an errand even though it's daunting.
I read in an evidence-based parenting book that studies have shown babies need to hear about 15 minutes of talking every hour to reap the most benefits. That felt like a lot to me until I thought of it as three five-minute conversations. Do them during diaper changes and you're halfway there!
The Wonder Weeks is a solid book with good ideas for how to interact/play with a baby, which is not intuitive to me at least! It also helps explain why they melt down every few weeks as a baby, which helped me feel less hopeless.
Also, I had a hard time asking for help because I felt like I should be able to handle it. I'm starting to realize that's crap and our kids do better with more caregivers/supported parents. I scheduled family members to regularly come twice a week and a babysitter to come once a week to give a break (my "break" with the babysitter is working to pay for the sitter, ha, but it's good for my sanity!) Just wanted to throw that out there.
Good luck! It's so hard to deal with sleep deprivation!!!