Small government would start with the ridding the system of the majority of social programs designed to "give free stuff" to people. Welfare and the like. It would also abolish the federal education system, and put the states in charge of it.
You aren't the only one that feels that way and I'm not the only one that disagrees with you.
There's no real point in arguing these two positions because neither is intrinsically right or wrong. They are simply different *opinions* about the role of government.
I do think it's fairly extreme to include the school system with welfare. I would also assume that you would favor the elimination of social security and medicare as well.
Of course I (and many like me) have an entirely different opinion of government and that includes the idea that government should provide a social safety net. So to me things like social security, medicare, public schooling and even universal healthcare are things that the government should provide along with provisions for defense of the nation.
There is no fact you can state that *proves* my opinion to be wrong just as there is no fact that I can state that *proves* your opinion to be wrong. These are value judgements as to what governmental (i.e. common) resources should be spent on.
No one I know is a true anarchist that believes we should have no government and everyone believes that there is some level of taxes required for government to work. All we are really arguing about is what the level (and distribution) of taxes should be and what are the priorities on which those taxes should be spent. Generally the way it breaks down is that liberals believe that money should be used to help the least fortunate of our citizens and conservatives believe that money should primarily go to large corporations (particularly defense corporations) and to wage war.
I don't expect to convince conservatives that they are wrong and that liberals are right, just as they should not expect to convince me of the opposite.
What I would like to convince conservatives is that the liberal view of the role of government is not an intrinsically evil or stupid way of looking at things and that given the premise that the role of government includes providing a social safety net that things like Medicare, Social Security, Public Education and universal healthcare are as reasonable as the need for a standing army.