Let $E$ be an elliptic curve over a $p$-adic field $k$. Let $\varphi: E \to E'$ be an isogeny defined over $k$. Write minimal Weierstrass equations with integer coefficients for both curves. Write $\varphi$ with fractions of polynomials with integer coefficients and consider the reduction of that map on the reduced equations. It should be clear that $\varphi$ sends a singular point to singular point. Also it sends tangents defined over $k$ to tangents defined over $k$. From looking at the possible cases, using the dual isogeny if needed, one should be able to conclude that if $E$ has good, split multiplicative, non-split multiplicative or additive reduction then $E'$ has the same of these four types. (Of course, one better looks at the isogeny between Néron models, but I tried to keep it elementary here.)

It is much harder to know what the exact Kodaira type do, especially in the potentially good, additive cases. In the multiplicative case, the types will be $I_n$ and $I_{pn}$; in this order or in the other order.

The conductor $N$ is defined such that ord$(N) = 0$, $1$ or $\geq 2$ depending on whether $E$ has good, multiplicative or additive reduction. So two curves with the same conductor will share the same reduction among these three types. However the conductor does not know if the multiplicative reduction is split or non-split. For instance the curves 37a1 and 37b1 will illustrate that.