We tend to be a bit more casual in general here, so you might want to take what I say with a grain of salt.
Firstly I have a persona that I've built from a combination of things that I liked in other paramedics. Its not fake so much as it emphasises, to a great extent, parts of my personality that are more conducive to putting smiles on faces.
In short I hope that it conveys energy, a certain respect for blokey stoicism, supreme confidence and a good deal of humour. I also hope that it offsets my baby face, naturally formal standoffishness and tendency to ignore small talk. You might like to do a little reflection on what your interpersonal strengths and weaknesses are and what you might like to do to improve any weaknesses.
"G'day, I'm Melclin, and this is (INSERT PARTNER), before we crack on, whats your name mate/love/other familiarity depending on appropriateness?"
"I'm Tom, how are ya?"
"Better than you by the looks of it mate. What can I do for ya?"
"Well I've had this chest pain you see."
"Oh no worries, so how long has that been going on for? etc etc."
I'll take their pulse pretty much straight after the introductions. Just for a second or two while we're talking. Fast normal or slow and extremity temperature. Good info to get early on. Means you don't spend 10 mins talking to them before you put the monitor on them and find they're in VT. Bit of human contact helps too. For your elderly types who have fallen, I'll often just hold their hand during the whole interaction. Old ladies love it. We say that these pts need, "Fifteen minutely there-theres". Alternatively, a few carefully chosen swear words, tactfully inserted can really put people at ease. Skater with broken ankle, "Ahhh it hurts wahhhh". Me, "Mate stop your whinging, I'm ganna fill you full of morphine and you'll be :censored::censored::censored::censored: faced in a tick, so its your lucky day".
The initial casual interaction puts people at ease, but I find I have to follow up with more intelligent language. Less jovial. More jargon, but with explanations. Reminds them that while you can shoot the :censored::censored::censored::censored: with the plebs, you're also an expert in your field.
I don't take any :censored::censored::censored::censored:. I'll sternly reprimand anyone who pisses me off in what I call, "Nurse voice". Its something I've copied from nurses I know who have a great ability to be simultaneously authoritative and stern but without letting you question their compassion. Stern humour combined with a reminder that we're on their side and that it is very unappreciative to be rude to someone who is trying to help you.
Then this:
It also helps to tell them what to expect and how long it usually takes.
Many patients do not understand in the hospital, no news is good news.
They are afraid, do not know what is going on, and while an ambulance ride is a normal part of your day, it is usually not a normal part of theirs.
Especially older patients have a very negative association with ambulances. Often the last they saw of a friend or relative was them getting put in one.
It goes a long way to ask them if you can help with their concerns, like making sure the house is locked, lights turned on/off, etc. They often also worry about how much things cost.
While you may not know, try to avoid saying things like "it doesn't matter" or "it's worth it" because it trivializes their fears.
It does matter to them and it may not be worth the long term consequences.
Condition permitting ask them if they need to call somebody.
Keys, wallet, phone, lock the house, someone to take care of the pets, make a few phone calls. Brilliant. We spent an extra 20 mins on scene with a bloke once, organising respite care for his frail wife before we transported. This stuff really matters. You can't just ignore it.
Talking to pts about what to expect and generally educating them is important. I get so frustrated with how little they know about their condition so I like to explain things in quite a bit of detail. So many of my pts say things like I took a turn and had to go to hospital for 3 days and nobody ever told be what was wrong. I'll often print off a copy of their ECG, write a few things down and have them stick it on the fridge for next time, write notes to GPs/home care nurses etc. You know, all that pt advocate crap that we're supposed to do sometimes.