You say that you want "present-day technology at the start of the super-spending spree". I'm going to take this to mean basically our current world.
Wikipedia has a pretty decent article on the economy of the United States. It more or less starts out by stating that
The United States' GDP was estimated to be \$17.914 trillion as of Q2 2015.
Well, let's round that to \$18 trillion. (This is in line with another figure in the sidebar of the same article, stating \$18.125 trillion for 2015.) You want the government to spend \$1 trillion of that on a space program. Per year. Every year. For an indeterminate future.
For Fiscal Year 2014, real-world US public finances are revenues of \$3.0 trillion and expenses \$3.5 trillion. I don't have any specific number handy for the entire US space program costs (your question does however state \$17.5 billion for NASA, which is likely a significant fraction but not the entire space program costs). We can probably safely assume that no matter how you slice it, the total cost is nowhere near a trillion dollars annually. Hence, we can simplify and say that you want to add an expense of \$1 trillion per year.
That brings the expenses to \$4.5 trillion. If revenues are left unchanged, this means that the budget deficit triples, from \$500 billion to \$1.5 trillion.
Individual income taxes, social insurance and corporate taxes add up to 90.7% of US tax revenues in FY2014. So these are the ones that you would probably need to raise if you wanted to actually fund such a spending spree. Good luck selling to the general public nearly outright doubling essentially all personal and corporate tax rates. Even in the current climate about corporate taxes, I think that would be a pretty hard sell. Corporate taxes make up 10.6% of the total taxation, so even throwing your corporate friends by the wayside and riding on a wave of discontent with corporate taxation to raise corporate taxes won't get you very far (and would probably force many companies to either shut down, or move out of the country).
Or, you could take on debt to get the money to blast off into space (either internally in the country, or in the international bonds markets, or a combination of the two). US public debt is already at 102% of GDP, or \$18.1 trillion in Q1 2015. If you add a \$1 trillion deficit to the existing \$483 billion deficit, every year, with no other changes (particularly, I'm ignoring the cost of servicing the debt, which probably will make matters worse, not better, in real life), in ten years you have racked up a total public debt of \$33 trillion. With GDP growth at officially 2.1% per year, merely extrapolating that curve, after ten years GDP is up 23%, but the public debt on the other hand is up 82%! In these ten years, the debt-to-GDP ratio grows from roughly parity to 148%. (And parity is bad enough, objectively speaking.) Compare this to Greece's 126% debt-to-GDP ratio per 2014 (and that's after some of the recent debt write-downs; it was 161% in 2012, when the country was deep into their financial dire straits).
Obviously, none of this happens in a vacuum (no pun intended), but your politicians will have to either sell it to the general public that they need to increase the total taxation by some 65-70% (which would bring the tax revenues from the current \$3.0 trillion to the almost \$5 trillion needed to balance the budget, assuming no dynamic effects which, again, are more likely to make matters worse than make them better), take on sufficient public debt that even with current GDP growth the United States would surpass present-day Greece in terms of debt to GDP ratio within a few years, or some combination of the two.
And this is before we even start considering the effect on things like wages throughout the society that your little project would have, which Thucydides so eloquently brought up.
Another way of looking at this is to compare it against the spending that went into the Apollo program. Apollo cost a grand total of \$25.4 billion in 1973. That would be somewhere on the order of \$150-\$200 billion today, accounting for inflation. If we are generous and call it \$200 billion, that's only 1/5 of your proposed spending, and over a much longer period of time. In other words, assuming the costs of Apollo were spread out over a decade, what you are proposing would be, in relative terms, somewhere on the order of 50 times the cost of the Apollo program. And you thought Apollo was expensive...
In summary: good luck. The politicians of your world are going to need it.