2003 Town & Country - Car will not start

When I put the key in & turn it, the panel lights up but I get nothing (no clicks,
etc.).  The battery is new and has the proper charge.  After several times
turning the key on-off-on, there appears to be a connection and the starter
engages for a moment then nothing again.  Eventually, the starter fully
engages and the car runs fine.  When I get to my destination and shut the
car off, sometimes it will start right back up and at other times, it will not and I
will have to start the process all over again.