FC sensors on or off?
 

I'm quite new to this game and have a question about using sensors like radar and active sonar.

Will using these sensors make my fleet easier to detect, like in real life? Or do I not have to worry about that?

This fare I have chosen to rely on Hawkeyes for long range detection, making shure to put the Hawkeyes some distance from the fleet.

They will. Those red lines that show up on your screen showing hostile emissions are also available to the enemy.

The instant you activate your radar or active sonar, you'll give away your position to every enemy for miles around, just like when Frodo put on the Ring and gave away his position to Sauron and the Nazgul..:)
You're doing good to use the Hawkeye's radar, the enemy will know it's there but if you hold it back out of SAM range they won't be able to touch it.
As regards when to switch on your ships radar, the general rule is to do it when you're pretty sure the enemy has already spotted you. An indication of this is when you pick up his purple ESM lines which means he's got his radar on and has probably detected you.
The downside of keeping your radars off for too long is that you won't detect incoming missiles until they're very close in visual range and your SAM's and CIWS might not have much time to engage them.