Can we really travel through earth's core?
As Chris Mueller said, in short: it isn't, or at least highly infeasible. Projects to drill into the mantle, such as the Kola Superdeep Borehole, have all failed because drilling equipment can't withstand the heat at only ~15km deep. Even if we were to come up with some sort of cooling system that's able to cool to 6400km or 12800km deep (depending on whether you would drill from one side only or from both sides at the same time), pressure is the second barrier that holds us from traveling through the earth's core. 

According to Lide (2006) the pressure in the inner core is 330 to 360 GPa, at which iron becomes a solid even at the high temperatures in the core. If you could drill as far as the core you would have to build a device that's able to withstand that pressure, because if you can't, the material surrounding your well would immediately become liquid and fill the hole, if not shoot up your well towards the surface.

There are no physical walls between the layers of the Earth, only transition zones where temperature and pressure combinations lead to different behaviour of the materials. An example is the Mohorovičić discontinuity, or Moho, which is the boundary between crust and mantle, below which temperatures are high enough and at the same time the pressure is low enough so that rock becomes either liquid or at least a "flowing" solid. Similarly, at the boundary between the inner and outer core the pressure is so high that even at those temperatures the iron becomes a solid.

Lide, D.R., ed. (2006-2007). CRC Handbook of Chemistry and Physics (87th ed.). pp. j14–13.