My one tip in addition to what has been mentioned here is to always make sure you start with name, age, code status and allergies. To me those are the 4 most important things in report. Name and age to make sure it's the right patient, and my absolute biggest pet peeve is when a nurse can't tell me a patient's code status and allergies. If someone is giving me report and has no clue, I'll stop report and log in to the computer in the room to check. I've had patients code during bedside report before- it's not worth taking the chance.
Other things you absolutely should know: What the patient's current drips are, any testing/labs that are pending or will need to be drawn, current oxygen/IV access. And a personal pet peeve as an ICU nurse- if the patient is intubated I want to know their tube size, and what marking it's at and at what landmark. Bonus points if you're able to say what the last ABG was and what(if anything) had been changed since then. This saves me the trouble of having to get a stat chest X-ray on a patient who shows up from the ER with an ETT at 19 at the lip, or having to dig through the chart to see if an ABG with a huge negative base excess had ever been addressed.
I want to be able to take over where you left off, not have to piece together what you did already as I go. I will say I'm saying all of this as a nurse that would get report from the ED, not an ED to ED report.