If you are flying domestically, many airlines offer Y-UP fares that book in coach but give you seats in first class. They are still expensive but may work for you. On AA, they call them "instant upgrade."
As already noted, you can upgrade coach to FC using miles. But wasn't noted is that you can buy miles from the airlines. On AA, you need 15,000 miles each way to upgrade, plus $75. You can buy those miles for $442.50. On AA, even the least expensive coach ticket can be upgraded, so this may be a cost effective alternative.
Best of luck -- Suzanne