If I travel in a straight line across the universe, will I end up where I started?

By travel in a 'straight line' you actually mean álong a geodesic, analogous to a jet plane flying along a great circle between two places on earth. Picking a direction defines the geodesic, and indeed you would come back to the same place (ignoring possible effects of inflation and issues of local space-time curvature arising from stars and galaxies passed). It may not really be a realistic question given these other effects, but conceptually just as would happen on the surface of the earth you would return to the starting place and traveling in the same direction. Don't expect to see the earth there; both it and the sun would be long gone (save perhaps for a tiny remnant of the sun no larger than the earth and frozen solid!).