If I make some software and start selling it, and then someone who has a patent on a central idea in my software discovers this, would they have to provide me a warning to remove my product from the market, or could they sue me directly without any warning, asking for a large amount of money?
They can sue you without any prior notice. The sum they can sue you for is determined by marketplace facts and a few other matters. If you are a solo programmer with a few hundred or a few thousand sales at a nominal price, say under $100 per unit, then there is not a lot to be had, so there is little incentive for the patent holder to take you to court without first warning you to cease and desist your infringement.
The main reason that patent holders are disinclined to give prior notice of infringement is that once you are warned, you have the option to bring your own law suit for a declaratory judgment (non-infringement and/or invalidity) in a federal court of your choosing. Patent holders prefer to pick the jurisdiction to suit their own purposes to the extent they can. The patent holder may be inclined to put you on notice by bringing a suit that they expect to settle immediately.
There is also the issue of whether, without any "prior notice" from the patent holder, you were already aware of the patent and chose to ignore it. In those circumstances you could be liable as a blatant or "willful" infringer. If you are found by a judge or jury to be a willful infringer, the judge may award up to three times the actual damages proved by the patent holder at trial. Note that he is not required to impose treble damages, but he has the option of awarding damages between the determined amount and three times the determined amount.
Other possibilities spin out from these basic considerations. Is a major corporation among your customers? As users of an infringing product, they are infringers. Perhaps the patent holder could go after them - even though it might be hard to show actual damages of any significant size.
But back to the original thought: If you are a little guy without much in the way of resources or money, you can probably get out of a law suit by withdrawing your product from the market and paying a more modest settlement. There are some circumstances when it pays to be penurious. This is one of them.