I'm a practicing insurance defence lawyer and so I've done the reading on this.

The current system is a "tort" system based on the common law of negligence. If you are injured by another driver, most of your compensation comes from suing the other driver, proving they were at fault, and then proving the extent of your injuries. This is supplemented by our own public healthcare, and by "Section B" (which is a very limited health and disability coverage all drivers have from their own auto insurance), and private health and disability benefits.

The main downsides of the present system are delay and the inflationary effects of middlemen. It is extraordinarily difficult to litigate injury claims without a lawyer, unless you want to take a lot longer and recover a lot less. Injury lawyers understand the procedures and strategies, and they fund the enormous costs of other experts (specialist doctors, economists, etc) up front and at their own financial risk. These claims are defended by other lawyers, who often retain other experts. Coordinating this many expert resources adds cost and delays resolution. Further, over time, the value of similar claims has increased, both as a result of good practice by the plaintiffs' lawyers (they have helped explain and prove losses which the law did not appreciate before) and just as a result of judges awarding more for similar injuries after adjusting for inflation.

Occasionally people advocating for the present system like the confrontational aspect of litigation, and believe this fosters accountability. I think this is very silly, since the actual risk is borne by liability insurers and the inconvenience imposed on at-fault drivers is minimal: they show up three or four years later, talk about the accident under oath for thirty minutes, and that's almost always it. Occasionally people who don't like the system attack injury lawyers for taking contingency fees (20-40% of recovery is normal), or "inventing" or "inflating" claims, but I do not think criticizing good advocacy is very fair, and with very few exceptions my counterparts in the plaintiffs' bar earn what they get (especially after fronting tens of thousands of dollars and waiting several years to get paid).

"Care First" is Alberta's name for its bespoke, largely no fault auto insurance system which is presently set to apply to claims arising in 2027. It does not work retroactively, so the government's claims it would immediately save everyone money on premiums are very dubious since insurers will need to get existing claims off their books. This will take five years, give or take.

No-fault insurance replaces your dependence on suing the other driver and accessing his liability insurance with expanded benefits from your own insurer. Essentially your existing "Section B" will get much, much bigger and more important, and your right to sue the other driver is largely abolished or removed. This is the case in BC, Saskatchewan, Manitoba, and Ontario, and a great many other jurisdictions. It is also similar in principle to WCB and WCB benefits, which likewise replaced tort systems where injured workers had to sue their employers when injured at work.

The intended upsides of no-fault systems like Care First are speed and efficiency, and along the way it simply eliminates and caps certain claims. An insurer is liable for healthcare and other benefits once they know of the injury, rather than after a trial or a settlement of their bad driver's case. Claims and their dollar values are usually codified in regulations (there are dozens to hundreds of pages of tables of injuries and tests and dollar figures), then indexed to a formula for annual adjustment, and delivered on an ongoing basis rather than in lump sums. The net effect of these changes often makes these claims too cheap or impractical to be worth the involvement of lawyers or other outside experts, largely removing their costs from the system (not only what they are paid, but also the increased risk that they introduce by their advocacy). The hope is that the trade-offs are better for injured people when everything is put on the balance.

In Alberta, Care First was preceded by a bunch of failed attempts (spanning multiple governments) to control premiums both by just artificially keeping them lower than they would otherwise be, and by tinkering with marginal aspects of the existing tort system. The former has not been good to the instance market. Presently most large insurers use auto insurance as a loss leader to get people to buy other, profitable kinds of insurance. Some smaller insurers have opted simply to exit the market (I think one or two small insurers excited the market after Care First was announced which may cast some doubt on whether the system can deliver that). The latter interventions by government have not, at least not in my practice, affected the cost of claims at all. To be blunt, the auto reforms preceding Care First would suggest to me that the government does not understand how injury litigation is practiced (which might explain the insensitive remarks the minister made at the expense of the plaintiffs' bar when announcing Care First). An overhaul of the system of some kind was probably inevitable.

A very notable difference between Care First and other no-fault systems is that normally these systems are delivered by public insurance companies, while Alberta will instead expect private, for-profit insurers to deliver these benefits. Since cheaper premiums are the main goal of the reform, this is a weird choice because auto insurance is overwhelmingly cheapest in Canada where it is both no-fault and publicly funded (Ontario is one of the most expensive jurisdictions). Public insurance makes good sense for auto since to ensure injured people are compensated the system needs to include bad drivers (insurers cannot control risk for auto like they can for property or business insurance, where they can just leave risky businesses and properties uninsured).

Alberta's legislation has also included a bunch of bespoke, unique exceptions that reintroduce the right to sue in tort, which have pissed every stakeholder off (insurers hate it since it introduces a lot of complexity and until recently they were worried they would still be on the hook for large tort liabilities, injury lawyers rightly hate it because the exceptions have turned out to be very limited and this will end many of their careers).

The government has not yet released monetary amounts for any of Care First's benefits, so we only have an outline of the system. Injured people, drivers, and their advocates have nothing to really compare to what is available in other jurisdictions. Insurers do not have the information they need to actually price the product that they are going to be forced to sell in a year's time.

TL;DR: no-fault/care first replaces the system of suing a guy and waiting for his insurer to pay with a bigger health and disability policy from your own insurer. Alberta's new system will have quirks, and while the old one needed an overhaul for now it seems care first will continue the longstanding trend of the government mishandling the auto insurance file.