Lot of concepts here.
1) 0% may feel like a threshold, but it's not that quantitatively different from the sub 2% interest rate environment we've been living in for many years from a wealth generation standpoint.
2) Should you stop saving? If you mean should you be generating more cash that you spend, the answer is that's probably a good idea unless you have a pressing cash need now. If by saving you mean putting your money in a Savings Account, then yeah there are probably better uses of your capital since most pay effectively 0% interest.
3) So where should you put it? This depends a lot on your tolerance for risk, and your forecast for how quickly you might need the cash. If you need it soon and/or have a low appetite for risk, then go for safer, less volatile assets like treasuries or CDs. If you have a bit of time and risk tolerance, stocks have pulled back considerably so buying in at depressed prices and waiting for the recovery might be a good idea. Real estate is probably similar depending on where you live. If you really want to swing for the fences, go ahead and start a business, although ultimately there are a million factors that will determine your success before the interest rate environment will.
4) Should you take out debt to buy even more assets than you could with my own cash? Maybe, this all depends on your risk tolerance. If your investment decisions are good, you will reap even more returns. But if they are are bad, you will have to pay back the debt after incurring losses. So it just pushes your outcomes towards the extremes. You probably should not do this solely because interest rates are low.
A final note-- your approach to wealth generation shouldn't change based on interest rates. I think your upbringing is mostly correct, if you work diligently and spend less than you make, you will be on a road to building wealth.
The question is what to invest your free cash flow in, and the answer is almost always: all of these options. Diversifying is the best way to minimize unsystemic investment risk.
The question is then: how do you allocate your capital between risky and safe assets? That depends on the interest rates, your near to medium term cash requirements, your risk tolerance, and your age. That said, a rule of thumb is to do (100 - age)% in riskier investments, and your age% in safer investments.