Which is the best time and way to buy US stock on US stock exchange on Tiger Broker?

Generally, pre and post market will experience low trading volume so you might find it hard to get a fill during those times. And especially for lower volume stocks (generally smaller cap), you might find that the price moves quite wildly during pre and post due to this lack of volume.

Hence, for most beginners, I'd advise you just buy in the regular trading hours. I tend to find that the opening few hours (until around 12) tends to have more volume and that's generally when the big moves occur.

I buy through limit orders yes. I like to have full control over the price I buy at. That said, there are also many who aren't too particular and just go for a market order so they don't have to wait for it to fill. This is an ok strategy too since if you have a long time horizon, a difference in like 0.5% probably wouldn't make much difference anyways.