Changes in which factors affect both the rate and the rate constant of a first order reaction?
If you see the rate equation for the Arrhenius equation:
$$
k = A e^{-E_a/(R T)}
$$
you can see that the rate constant increases for an increase in temperature (and as activation energy increases, the rate constant decreases.

Now, since the rate constant increases, this implies that the rate of reaction increases with temperature.

Regarding concentration: you are correct in that concentration affects the rate of reaction. However, a rate constant does not change according to concentration. You can observe this in the Arrhenius equation, where A is independent of the concentration of the substrate. The fact that the rate increases does not imply the rate constant changes.

Possibly more clearly: it is implied that increasing the rate constant will increase the rate of reaction. However, increasing the rate does not increase the rate constant.

An increase in temperature increases the rate constant and hence the rate.

An increase in concentration increases the rate but not the rate constant.