I agree... start with a college-level A&P program and I don't mean "Airframe & Powerplant" program either.
By taking an Anatomy & Physiology course of study (look at your local RN program as you'll want the same level A&P they require) you'll gain a much greater understanding of how the body works and why so that when you're studying the abnormal stuff in your Paramedic program, you'll understand the
why behind the abnormal stuff. This may not seem like much now but you'll start understanding this later on when you are much more quickly able to grasp the significance of some subtle sign you're seeing.
Yes, I'm an RN. This does NOT give me greater abilities than I had when I was a Paramedic... I have the ability to use different viewpoints into patient care than I used to but assessment is assessment regardless of level. It's up to the clinician to be able to make heads or tails of the data gathered. What I do have that Paramedics normally don't is a LOT of backup right at my fingertips. Usually the providers order labs or have protocols that cover lab work so that I have that available to me as well. One last thing that I am developing that Paramedics rarely get to is an ability to prioritize multiple patients. It's the norm for me to have 3-4 patients at the same time, all shift long where as a Paramedic I usually had only one patient at a time.
As far as "picking up skills" go, you could start learning to read things like ECGs but it'll be tough to teach yourself this stuff. Most courses in ECG interpretation (at least those that I've seen) concentrate on teaching you to recognize STEMI and the arrythmias and not much else. There's a whole lot more to reading an ECG than just looking for STEMI and doing rhythm interpretation.
What I would suggest you do more than anything (besides taking A&P) is focusing on sharpening your BLS skills. The reason for this is that the vast majority of patients you'll see as a medic will be at the EMT level. You want to sharpen your skills in moving the patient around. You need to learn all those operational things so that when you do become a Paramedic Student and later become a Paramedic you don't have to learn that stuff. You'll already know it like the back of your hand so you won't be fumbling about how to take a blood pressure while attempting to assess a patient that clearly needs some kind of intervention beyond that of mere hand-holding, positioning/pillows, oxygen, and splinting.