You're right that the reason she usually doesn't just turn over and go back to sleep on her own is because she goes to sleep in your arms.
It is to do with the environment checks children (and adults) do in the brief moments between sleep cycles. Imagine going to bed one night and rolling over a few hours later and realising upu were in the backseat of a car. You wouldn't just roll over and go back to sleep, you'd be wide awake adrenaline pumping pretty fast. It's the same process when baby goes to sleep in your arms then in the environment check is no longer asleep in your arms.
As for what to tackle first - is most suggest both bedtime and daytime naps at the same time. This allows for the greatest levels of consistancy.
However if you can't face doing both, then it would be most wise to do bedtime and night wakes first, leaving daytime naps to tackle later. The natural pressure on the body to sleep is greatest at night so sleep routines easiest to establish.
If you do decide to teach her to go to sleep in the cot, the key is consistency. Your child simply will not stay awake for all 12 hours over night, so you just keep going with in-cot settling from awake until she eventually does calm down and settle.
I wouldn't leave her to cry, i would stay in the room. Keep lying her back down, shushing, stroking, patting, whatever is needed to get her to calm down whilst in the cot.
Your initial aim is not sleep, it is staying lying down. So keep in lying her back down over and over again. Shush, stroke, pay and calm her until she stop pulling herself up. The next aim is calm, so still your movements but stay close and touching, pat and shush if not calm, then still and wait once calm. After calm and lying down; your next aim is sleep. No point starting the whole process expecting her to go to sleep just like that. Certain conditions are needed first: lying down, still, calm, quiet.... only then can come the state of relaxing and sleeping. So don't rush, baby steps.
The first night it might take several hours. But you stay comforting throughout. It should get quicker and easier each day.