What does a chest sound like?
Sorry, not trying to be a **** but if we want to be recognized as a medical profession we need to use proper terminology such as lung sounds and heart tones.
Chest sounds = lung sounds and breath sounds anteriorly. It is a generally accepted term, at least at my college of medicine. Should you use it in a medical journal? Probably not, but it's perfect for day-to-day use in oral presentations or on an online forum.
But to the point of the OP.
For everyone:
10 second survey. Is this patient sick or not sick? Do we have evidence of Airway, Breathing, Circulation, Disability? This is going to be the hardest part of "vitals" and just comes with time. No one can really train you on how to do a 10 second survey.
After that, for a stable patient:
1) Talk to the patient and make introductions to assess cognitive function I don't need a full GCS number, but I want an estimate. I do this while reaching in the jump bag for equipment I need.
2) Place Pulse-ox on a finger contralateral to the side I will take a BP on. Let it sit for a second to capture. Great time to do a cap refill too.
3) Place BP cuff (or as Robb might insist we call it, the sphygmomanometer

) on the patient.
4) Palpate radial pulse to ensure it matches the waveform/light pulse on the pulseox. If it does, then I have my pulse reading. If not, I'll have to get bilateral pulses later.
5) Take BP
6) Take respiratory rate. I usually do this while writing down other things to not make it obvious, or during the manual pulse if the pulse-ox fails.
7) As needed, take the pulse rate manually
From here, depending on what the CC is, I'll look at breath sounds, pupils, glucose etc. in an order that makes sense for the CC.
After that, for an unstable patient:
Correct Airway, Breathing, and Circulation in that order. Only check vitals if it's really important to something I'm doing (such as a contraindication for an intervention) and even then, only check relevant vitals.
On unstable patients, there have been transports where I never took one set of vitals since in the time it took to get from the scene to the hospital, I never got past A, B and/or C.
Some would argue that stable patients should always get a full set of vitals. I disagree. If I'm 2 blocks from the hospital where my driver can have me to the bay in 30 seconds, I'm not going to sit on scene for 5 minutes just to get a complete set of vitals and history and all of that. For example, in an MVC 4 blocks over from the hosptal where the patient self-extricated but wants to go to the hospital, my order might very well be:
1) 10 second survey--he's stable
2) C-spine clearance protocol - I can clear, hop on into the truck for me please!
3) History--what happened. Weren't paying attention and ran into the pole? Or did you get dizzy and pass out first?
4) Courtesy Call to hospital to let them know we are inbound, no entry note, just a call only because
5) We pull into the ambulance bay.
Sitting on scene and taking a full set of vitals and a full history when you can see the hospital is the BLS equivalent of sitting on scene and trying to get an IV for 20 minutes. Just because you haven't done much doesn't mean you haven't done the best possible for the patient.
A patient having an MI doesn't need an IV--they need a cath lab. The IV is part of the requirements for a cath, but your goal is cath, not IV. If you can't get it, you can't get it, move on and let the hospital get the IV where they have ultrasound and other cool toys.
Vitals are just that--a tool for assessing and treating the patient. But they are not the end goal.