I suppose your need for a rental car has already come and gone, but I’ll toss my hat of knowledge into the ring for your future rentals or for others who have the same question.
The first thing to note: you mention being new at driving. It’s important to note that most car rental companies have age limitations. Few will rent to persons under 21, and most will charge a fairly large fee ($10-15/day) if you’re under 25 years of age. Be sure you’ll even be able to rent a car before committing to plans that need one. (If you’re over 20 years old and are new at driving, then you must have grown up in NYC. ;))
The second thing to understand is the difference between collision and liability. It sounds like you do, since you specified that your American Express does not cover it, but many people don’t really understand the difference. For them: collision coverage covers the car you’re driving. Liability protects you against a claim (or suit) from someone else you’ve hit–they of course want you to pay for their property damage (car repair) and bodily injury (medical) bills.
Now, most credit cards issued in the U.S. cover some form of the former (collision), but none (literally, zero) cover third-party liability. With collision, there are some exclusions, and the hoops you have to jump through to make sure your credit card will cover you can be fairly onerous. I would never suggest relying on a credit card as your only source of coverage–I’ve seen (first-hand) too many horror stories of coverage being denied on a technicality and people having to pay thousands of dollars out of pocket for damage to a rental car. Credit cards are fine to rely on for secondary coverage–covering a deductible you could, in a worst-case, afford to pay on your own if your credit card claim is denied–but I would not recommend chancing having to pay tens of thousands of dollars out of pocket if they decide not to pay your claim. Traditional insurance companies with real comp/collision coverage are much better to rely on (and there are even some legal opinions in Maryland that obligate insurance companies to provide this coverage–see Page on state.md.us).
For liability, the answer is a little more complicated. Assuming you’re renting in Maryland (not D.C. or somewhere else outside the state), state law requires the rental company to provide primary liability insurance but only to state minimum limits (30/60/15–source: Insurance Requirements for Maryland Vehicles), which are not high enough to adequately protect you in any but the smallest of accidents. For more on this topic, see my other answer here: Chris Luth’s answer to What’s the best car rental insurance option for someone without any coverage?
So, what are the best options? If you’re only going to be renting this one weekend and perhaps a couple of other weekends throughout the year, the simplest and likely least expensive thing to do is simply to purchase both CDW and supplemental liability insurance from the rental company. The total cost for these two products will probably run you about $35 a day, give or take $10, but in exchange for this, you’ll get $1 million in third-party liability (well more than enough protection for you) and zero-deductible collision damage waiver. (Yes, your American Express wouldn’t cover you, but that’s no matter since you would not be responsible for any damage to the rental car anyway.)
If, however, you forecast renting for more than a week or so over the coming year, you’ll want to look into other options. Unfortunately, since you (presumably, since you’re asking this question) don’t own your own car, your options are limited. You need to specifically look for “non-owner auto insurance.” Policies like this are not common, but as I mentioned in my other Quora answer I linked above, you may have luck with GEICO, AAA, or, if you qualify, USAA. Progressive may offer it, too. These policies might only cover liability, though, so you’d still need to consider collision coverage. One other option is to purchase CDW through a third-party like this one: Rental Car Insurance – Travel Guard–it’s cheaper than the rental company’s own CDW, and although it’s not a regulated product like traditional auto insurance, at least you’d have two sources of coverage (that and your credit card), which I would feel comfortable relying on.
Last, to address your question about the legal aspects of driving without insurance: I should of course clarify that I am not a lawyer. However, while yes, in most cases I’m aware of, it is illegal to drive without third-party liability insurance, note that pretty much everywhere outside of California, if you rent a car, you won’t be driving without third-party liability insurance, since every other state requires rental companies to include at least state-minimum coverage, either on a primary or secondary basis. That coverage might not be enough to adequately protect you, but it is enough to drive the car without being concerned about being on the wrong side of the law.
I’m a big advocate of carrying adequate protection because I’ve seen (and experienced myself) many cases where the unexpected happened. Others (and yourself) might be more cavalier about it, but I applaud your concern with being responsible and as well-protected as is reasonable and possible for someone in your situation. Enjoy your weekend rental and hopefully, your introduction into the world of rental cars goes smoothly!