There is the Fleet Carrier's Owners Club somewhere on Discord and Reddit, I believe, where CMDRs organize carrier trips. You could go there. Carrier trips also count for the 5k trip for Professor Palin. If you take a carrier somewhere to explore you still need to travel away a bit, as most of the carrier trips will probably have some more or less popular destination.So if exobiology is good how do I get on a carrier that's going out deep in space and then what about the return trip ? I suppose 2nd option is to get some extra LYs travel on the neutron star path, but it does damage the ship a little. then you have to get home. CC
If you want to do a trip by yourself, go in a direction that is not heavily traveled, preferrably somewhat outside and away from the galactic plane. You could take a look at EDAstro to get a feel for where to go. In general, the straight routes to Colonia, SagA* and the nebulas surrounding the bubble are well traveled.