Congratulations!
You're right -- at 10 days she's too young to cry it out. You should respond to her cries as soon as possible (I wouldn't let her cry more than 10 minutes at a time).
Also, babies tend to go through a growth spurt at 10 days, so she may be more hungry than usual. Otherwise, just feed her when she's hungry, hold her lots and sleep when you can. You don't have to worry about sleep training until she's 10 or 12 weeks. By then, she may have done it on your own and save you lots of trouble!
She will naturally start sleeping longer and straighten out her days and nights on her own too. The main thing you might focus on is simply the routine of Sleep, Eat, Awake. Feed her when she wakes up, try to keep her awake for 5 or 10 min after a feeding (this can easily be accomplished with a diaper change -- cold wipes should wake her up every time), then let her sleep as long as she wants.
You're instincts are right about not keeping her awake. Let her sleep! She's got a lot of growing to do! And I wouldn't wear yourself out trying to wake her up at certain times either because as you've already noticed, it's next to impossible!
You really won't get a lot of sleep in a row for a few weeks, so get it when you can. (I remember feeling of that glorious moment when we went to sleep and didn't wake up for 4 straight hours.) She should be eating every 2 to 3 hours (from the beginning of one feeding to the beginning of the next), so that doesn't leave much time for rest except in one or two-hour increments throughout the day. There will be some point during the day when she naturally sleeps a little longer. Take advantage of it and lay down!
If she's sleeping in the early evening, lay down then. I remember when our first child would sleep her long stretch between about 8pm and midnight/1am. We were not going to bed until 10pm, only to be awakened a short time later. So we tried to go to bed when she did. It was weird getting in bed at 8 or 9pm, but we were so exhausted, all we had to do was sit still a minute or two and we'd fall asleep!
My second child would naturally sleep a long time in the middle of the day -- like noon to 5. When I realized this was a pattern, I started napping then too. The point is, get it when you can! Dishes can wait, tell visitors to go away, turn off the phone and get some sleep!
For getting the baby to sleep, each of our kids were different in the beginning, but we would hold them until they were almost asleep (that point when their eyes close and their head bobs, but they're not completely asleep), then lay them down.
For my daughter, I'd swaddle her and put her down and she'd fuss a little, so I decided I would come back to comfort her every 10 minutes. I literally watched the clock (from another room, of course). At first, she would fuss the whole 10 minutes, but would go to sleep within 5 minutes of the first time I went back to her room. We never made it to 20 minutes. After a while, she never even made it to the first 10 minute mark. I'd go back to find her sleeping soundly. Sometime by the end of her first month she got pretty good about going right back to sleep after middle-of-the-night feedings too.
My second child was a different story. He spent the first 2 weeks of his life sleeping on my chest in a recliner because he did just like you are describing your baby.
With a toddler in the house we were more desperate for sleep than the first time around, and he would kick out of even the tightest swaddle, so I held him til he went to sleep, then put him down on his tummy. Our problem with getting him to stay asleep was solved. Yes, we felt all kinds of bad, but nothing was in his bassinet but the fitted sheet and the baby. I would certainly try other methods before tummy-sleeping, because the AAP has good reason to recommend back sleeping.
Of course we did eventually have to get him to go to sleep without being held, so around 6 weeks or so we did the same routine as our first -- hold til he's almost asleep, then put him down, then he would scream like a banshee as we walked out of the room. We'd come back every 10 minutes. It would often take 3 visits before he'd go to sleep. It was exhausting to listen to the crying and to have to keep going in there. So one day I decided to just wait a full 20 min. He went to sleep at the 18-minute mark. He did that every time for days and days. After a few weeks it got less and less until it was less than 10 minutes.
I had read somewhere that some kids are comforted by getting "checked on" and others are actually made more upset. Apparently I had one of each of those kinds of kids!
For swaddling, I've heard great things about the Miracle Blanket (maybe if we'd had this we wouldn't have put our son on his tummy!).
Good luck!