Main options are:
a) a taxi cab - its actually a fixed rate between LAX and downtown LA of around $47, plus a tip for the driver
b) a shared van shuttle, like SuperShuttle (www.supershuttle.com) or Prime Time Shuttle (www.primetimeshuttle.com) - around $30 for the two of you
c) take the FlyAway bus from LAX to Union Station in downtown ($7 per person) and then take the subway (Red or Purple Line) from Union Station one stop to the Civic Center station ($1.50 per person) - its about a 3 block walk from the subway to the Omni.
d) Take a private car or Limo - Prime Time Shuttle, for instance will do this trip in a private town car for around $88.
Bottom line, a taxi is going to be more expensive than the shuttle for only two of you - but the shuttle can circle the airport to try and fill the van, and can detour along the way to pick up or drop off others. The taxi takes you straight there. The FlyAway / subway combo is the cheapest but you'll be doing some walking and not sure this is the best idea in downtown LA that late at night on your first night in LA. Private car / limo most expensive but also the nicest way to go.