The US government started printing paper money to fund the Civil War... How come this didnt cause I'm nflation?

Well, the short answer is that this did cause inflation. In Tariffs, blockades, and inflation : the economics of the Civil War by Mark Thornton and Robert B. Ekelund they have a good summary of the effects of inflation both north and south during the war; and while we normally focus on the hyperinflation of the Confederacy, the United States also had to deal with inflation from printing greenbacks. According to one study, quoted in Thornton & Ekelund on page 71, $100 in 1860 was worth $175 in 1865, reflecting the increase in the money supply provided by the Legal Tender Acts.

Before the civil war, the federal governmentbused gold and silver coins. Was this the other onlu currency?

Before the Legal Tender Act of 1862, the only legal tender in the United States were gold and silver coins, which the government was empowered to "coin money" under Article 1, Section 8 of the Constitution. However, this was not technically the only "money" in circulation; in 1860 the United States was in the "free banking" era, when banks (and other entities, like state governments and railroad corporations) were unregulated and could offer their own private currency - these were paper promissory notes generally unbacked by specie and not legal tender. There were hundreds, if not thousands of these private currencies in operation throughout the United States. There's a great example of how this actually worked, quoted in Hugh Rockoff's Lessons from the American Experience with Free Banking:

M.E. Ames, a St. Paul attorney, reported his experience on April 11 and 16, 1859. He said he had received $200 of Owatonna notes from a 'respectable Banker of this City'. (Ames did not say, however, what he paid for these notes, so that it cannot be assumed that he purchased them at par.) Trying, then, to purchase $125 of exchange on New York (New York bank notes), he said he could not find a bank that would make such a trade 'at any price.' Since New York exchange was roughly selling at par, the implication is that Owatonna money was selling for less than 63 cents on the dollar. [Ames eventually took his notes to Pease who, as Ames knew, was the Owatonna broker. With some difficulty, Ames persuaded Pease to buy the notes for St. Paul city scrip, but could get neither New York exchange or gold out of Pease.]

So many folks in the United States would have been familiar with paper money even before the Legal Tender Act of 1862; specie (gold and silver coins) were preferred, but these too were famously variable in price, due in part to the increased supply of gold due to the Gold Rush in California, and because of the sinking of the SS Central America with its shipment of gold in 1857 (the US Congress also passed the Coinage Act in 1857, making foreign coins no longer legal tender, because the Spanish silver dollar will still in relatively widespread use).

Still, the road to the greenback wasn't easy or automatic - the US government tried several different ways to raise money for the war, including issues of Demand Notes in 1861, which were redeemable in gold (at least initially) and later interest-bearing, but weren't legal tender--for a full look at all the different variations on US currency, you'd want to look at Paper money of the United States : a complete illustrated guide with valuations. The idea for a paper currency unbacked by gold was, according to Abraham Lincoln floated by Col. Edmund "Dick" Taylor (I've seen another version of this quote floating about, but I've never been able to track down a source for it).

The use of these notes as legal tender (which meant, among other things, you could pay your taxes with it) and their use to pay soldiers, combined with the National Banking Acts of 1863 and 1864 (which ended the "free banking" era and significantly reduced the number of private monies in circulation) pretty much began the transition of the population of the US to adopt greenbacks over specie money - although it was a long battle, and it took the Legal Tender Cases decided by the Supreme Court to authorize that paper money as legal tender was, in fact, authorized by the Constitution.