Hugs to all of you struggling with those early weeks of sleep frustrations!! Just wanted to mention a couple of things that really helped with my DS when he was just a few weeks old -
a super consistent wind down routine is essential. Ours was, walk into room, swaddle, turn lights off and sit in chair with baby hugged into my chest, pat back and shhh by baby's ear until they start to relax and eyes get droopy (usually about 5mins), saying 'it's time to go to sleep'. Then lower baby into bed (if it's not hot, then preheating bed with hoer bottle really helps them relax into bed). Tuck in snug and give couple of pats, then leave room. If he then got upset and couldn't settle, I'd go back in and do pat/shh (or pick him up and re-calm him if he was really crying hard).
At about 8 weeks we had to put DS into bed before he started to get sleepy/eyes closed cos he'd start to jolt awake as soon as he hit the bed. We tweaked the routine to involve singing 10green bottles before putting him into bed still wide awake. By then we also had a little thing that made a noise like ocean waves for 20mins after we put him to bed (it's a bit like someone going shh...shhh...shhh), and a little musical mobile that goes for a couple of minutes. All of this isn't so much to soothe him to sleep as to cue him that he is about to go to sleep. It takes a few days of consistently using your routine for all naps and bedtime before you'll see the results.
Something to keep in mind when doing that frustrating pat/shh is that you and baby are on the same side - they genuinely want to sleep, they're not actually fighting it, it's just that they don't know how to go about making that transition- especially if they've always been rocked or fed to sleep. I find it really helps to keep my sanity by gently repeating 'I know you're so tired sweetie, we're just teaching you how to get to sleep and then you'll feel so much better...'.
When DS was little the yawn was the first obvious sign that he was getting tired (and perhaps the eye rubbing but since they can't control their hands at that age, it was often just a general, tight fisted face rubbing), I acted on that first yawn to begin the winddown routine (usually at 50mins after his last sleep). If your winddown routine is a bit longer then you'd want to start before those first signs.
Also even though it's hard to get it going, the sooner you get LO into the EASY routine, the better, because sleeptime is more predictable for both you and LO - they like to know what's going to happen next. So they'll be more prepared to go to sleep when the time comes. It's tempting to leave it till they're older rather than face a struggle now, but unfortunately teaching anything just gets harder as they get older (and bad habits get more ingrained and harder to break - like rocking and feeding to sleep) so while it might take you a week to get it going now, it could take two weeks and a lot more crying later on.
Anyway, I promise that things will get better as LOs get older and more settled into routine - you'll be amazed at the difference easy can make to your enjoyment of the parenting thing. It's not always smooth sailing, there are hiccups along the way, but once back on track it can work a treat for both of you.