“We have to live today by what truth we can get today, and be ready tomorrow to call it falsehood.”
—William James
It sounds to me like what you’re really worried about are the fundamental questions of epistemology: What is knowledge? How is knowledge acquired? To what extent is it possible for a given subject or entity to be known? These are theoretically prior to questions of science. The skeptical position you mention—i.e., that we know only the contents of our own minds—is known as solipsism.
In answering some of your questions, we first need to be aware of the difference between truth and facts. Facts are a states of affairs that obtain in the world. Truth is a property that a certain kind of sentence might have. If you have a pet bird and it is perched in its cage, that the bird is perched in its cage is a fact. Furthermore, for as it remains a fact that the bird is perched in its cage, the sentence “the bird is perched in its cage” is true. Note, then, that truths reference facts and are true in virtue of certain facts obtaining.
Before we try to figure out which things are true and which possible states of affairs are actual (and thus factual), we need to address the question you ask about how we know that there are such things as facts and truths in the first place. The answer is that the alternative cannot possibly be the case. If there were no facts, then it would be a fact that there were no facts. Thus there would be at least one fact, meaning that it would not be a fact that there were no facts. The same argument can be given with regard to truths. If there were no true sentences, then the sentence “there are no true sentences” would be true. Yet this would mean that there was at least one true sentence, so it would not be true to say “there are no true sentences.”
A similar kind of strategy works for knowledge. Some people would argue that 100% certainty is required for knowledge. If you’re evidence does not absolutely guarantee that a particular sentence is true or that some fact obtains, these people would say, then you cannot say you know that the sentence is true or that the fact obtains. People who argue this way are called infallibilists. They hold that the standard for knowledge is that you could not possibly incorrect. Infallibilism is frequently used as support for skepticism (where that term is understood as involving a rejection of all claims to knowledge). The skeptic asks “are you 100% certain?” When we say that we are not, he then asks “then how can you say that you know?”
This is an argument that many people find compelling, but it is a trick. The question for the infallibilist is this: how does he know that his standard for knowledge is correct? Does his evidence guarantee that only 100% certainty could give us knowledge? It does not, so his own standard fails to support his definition. As such, we need an alternative to the infallibilist’s criterion for knowledge. This alternative is called, perhaps unsurprisingly, fallibilism. According to the fallibilist, we are justified in calling something “knowledge” even when we do not have 100% certainty—though we must always be open to the possibility that future evidence may overturn our present convictions (this is the point of the William James quote placed at the top of this post).
There are a lot of things that could be said here about fallibilism. The most relevant point to make for the purposes of your question, though, is that science proceeds on the basis of fallibilism. Even where science is thought to have proven something, that proof must always be understood as having a footnote reading “pending further investigation.” The reason that people tend to accept the most current results of science, then, is that—as @nikipedia said—science works. The claims made by scientists are among the most rigorously investigated pieces of information that we have. This means that once we accept fallibilism over the (unworkable) infallibilist alternative, science represents one of the best ways of discovering what is most likely to be true. Thus it is well worth adhering to scientific results until they are overturned—at which point we adhere to newer, better confirmed scientific results.