Yes, you're correct. As the current passes through the primary coil, the flux linked with the coil changes, as a result of which a current is induced in the secondary coil.

Faraday's Second law of Electromagnetic Induction, states that The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux enclosed by the circuit, i.e. $$ \epsilon = - N \frac {d \Phi }{dt} $$ where N is the number of turns in the coil.

So, if the flux change is $d \Phi $ in time $dt $ then $$\frac {\epsilon _{Primary \; coil}}{\epsilon _ {Secondary \; coil}} = \frac {N _{Primary \; coil}}{N _ {Secondary \; coil}}$$

So, if the number of turns in the secondary coil is greater than that in the primary coil, the EMF induced, increases for same flux, while a lesser number of turns results in a smaller EMF.

The core is magnetised, that is, a flux change is induced, by the current in the primary coil. 

As the current passes through the primary coil, the flux linked with the coil changes, as a result of which a current is induced in the secondary coil.

Faraday's Second law of Electromagnetic Induction, states that The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux enclosed by the circuit, i.e. $$ \epsilon = - N \frac {d \Phi }{dt} $$ where N is the number of turns in the coil.

So, if the flux change is $d \Phi $ in time $dt $ then $$\frac {\epsilon _{Primary \; coil}}{\epsilon _ {Secondary \; coil}} = \frac {N _{Primary \; coil}}{N _ {Secondary \; coil}}$$

So, if the number of turns in the secondary coil is greater than that in the primary coil, the EMF induced in secondary coil increases (step-up transformer), while a lesser number of turns results in a smaller EMF (step-down transformer).