This may be completely irrelevant, since I am in NJ and you are not, and state laws vary tremendously.
Here, people (especially those of the male persuasion) under the age of 25 pay much higher insurance, since they tend as a group to have higher accident rates. After that, if one has proved to be a good driver throughout that risky time, yes, the rates will often drop considerably. If you were fortunate enough to be female and to have taken a driver’s education course, your rates even as a teenager would not be increased much at all since young ladies don’t, as a group, tend to have as high an accident rate. (It’s probably too late for you to do much about that now.)
The type of insurance required also varies from state to state, I believe. Hereabouts, what’s required is liability – insuring that any damage you cause with your vehicle will at least be somewhat compensated for. Here, the minimum is, IIRC, $15,000. The other options such as PIP, collision, etc., are not mandated but are often highly recommended depending on your assets, other insurance, and the value of your vehicle. Obviously a 21-year old who has other health insurance and who owns little or nothing besides the second hand clunker he got for $1500 has little reason to buy more than the minimum. Some of the peripheral items such as comprehensive, towing, etc. don’t cost that much anyway; the bulk is liability and collision.
This is complicated by whatever your state’s rules are regarding “no-fault.” That is, if you are hurt in an accident that is someone else’s fault, are your medical bills covered by your own or the other person’s insurance?
As far as the difference between owning and still paying for your car: I think what you’re talking about is that whoever you borrowed the money from insists that you carry collision insurance on the car since it is the collateral for the loan. Presumably (unless your state requires otherwise), you could drop the collision & other coverage after you pay off the loan. Whether you do or not depends, as above, on how much risk you are willing to assume. Your insurance company should be giving you a breakdown of how much of your premium goes for liability, collision, etc.
I hope this is helpful; again, your current insurance company should be able to give you more specific information appropriate to your circumstances.
