This objection of being skeptical about everything is rooted and grounded in philosophy of uncertainty. In fact it is not really all that new in that it has its roots began in ancient times before Christ. In a not so ancient time, like the eighteenth century, we would find a "champion" of this form of skepticism in the likes of David Hume (1711-1776).
David Hume was a Scottish philosopher who was one of the preeminent thinkers during the Enlightenment. Hume claimed that all knowledge of "mattes of fact" or any knowledge not grounded in meanings of terms, is based upon our senses. Within the discipline of religious thought, Hume became famous for his arguments that belief in miracles was irrational because the evidence of a past experience will always outweigh the testimony in favor of miracles.
So suffice it to say, this challenge, to the Christian worldview, of doubting everything falls within the context of our believing in God, and the existence of anything supernatural. Someone who is skeptical of anything supernatural would be "naturally" indecisive and seek to rip apart a Christian's views by "throwing down this objection. So is there an answer to "doubt everything that claims to be true." In fact there is. Let's move forward.
Understanding and Dismantling the 'Doubt' Bomb
Skepticism says that we need to hold off on the decision of following Jesus Christ. This threat to Christianity for indecision is really not too difficult to answer for three reasons. The first thing we need to understand is it is impossible to be skeptical about everything. To be skeptical about everything means that we would have to be skeptical of our own skepticism. That is not too comforting.
Secondly, when one is convinced to "hold off " on making a decision about the world, they are actually making a decision about the world. Many people fail to see that making no decision in essence is making a decision. Think of it this way.
Let's take a decision that of us might be able to relate to, our cars. Suppose your car is stopped on the railroad tracks with an oncoming train a few hundred feet away. Do you think it would be or not be wise to make a decision at that moment. If one in that situation made the decision, not to make a decision, would it be a wise decision? Of course not! If no decision was made to move the car out of the way of the oncoming train, the person has actually made the decision to stay in harm's way. That would not be a wise decision would it?
Thirdly, Jesus Christ combatted skepticism during His ministry. After His resurrection, He addressed His disciples' doubts by showing His pierced hands and feet for examination (See Luke 24:37-39 and John 20:26-29). Jesus further reduced their skepticism by eating a piece of broiled fish with them (Luke 24:42).
So having this understanding of skepticism in our modern day, the next time someone tells you that you should doubt everything that claims to be true, ask them this question: "Should I doubt that too?" One cannot doubt everything without being forced to doubt their own doubt.
 Allow me to point you to C. Stephen Evans' book, Pocket Dictionary of Apologetics & Philosophy, and check out the entry on agnosticism, skepticism and David Hume.