I'll answer the title question, which I will interpret as being about direct implications, that is, whether all $\theta$ cardinals are $\sigma$ cardinals. There are some general rules for which large cardinal properties imply each other depending on their [Lévy complexity](https://en.wikipedia.org/wiki/L%C3%A9vy_hierarchy):

 - If $\theta$ is weaker than $\sigma$, then, of course, $\theta$ does not imply $\sigma$.
 - If $\theta$ is stronger than $\sigma$ and the complexity of $\theta$ is less than or incomparable to that of $\sigma$, then $\theta$ generally does not imply $\sigma$. Exception: inaccessible cardinals are worldly even though inaccesibility is $\Pi_1$-definable and wordliness is only $\Delta_2$-definable (if wordly cardinals were $\Pi_1$-definable, worldliness would be downward absolute to inner models, and [it is not](http://jdh.hamkins.org/worldly-cardinals-are-not-always-downwards-absolute/)); I think there are other exceptions involving virtual large cardinals but I don't know them that well.
 - If $\theta$ is stronger than $\sigma$ and the complexity of $\theta$ is greater than or equal to that of $\sigma$ and at least $\Sigma_2$ or $\Pi_2$, then $\theta$ generally implies $\sigma$. Exception: Enhanced supercompact cardinals are not generally $C^{(2)}$-superstrong even though both enhanced supercompactness and $C^{(2)}$-superstrongness both have complexity $\Sigma_3$.
 - If $\theta$ is stronger than $\sigma$ and both have complexity $\Delta_2$ or $\Pi_1$, $\theta$ may or may not imply $\sigma$.

For $\Delta_2$- or $\Pi_1$-definable large cardinal properties, a finer hierarchy is useful. We can call it the extended local Lévy hierarchy and it is based on alternation of quantifiers over the elements of $V_{\kappa + n}$ (subsets of $V_{\kappa + n - 1}$), where quantifiers over lower ranks are treated as bounded quantifiers. Large cardinal properties generally don't imply other properties that are higher in this hierarchy.

Here's a list. Most of these properties are described in [Cantor's Attic](http://cantorsattic.info/Upper_attic).

I'll start with $\Delta_2$-definable properties. I'm not sure which of them are $\Pi_1$-definable but I note where I'm sure that they are. Additionally I note their complexity in the extended local Lévy hierarchy.

 - The weakest large cardinal property is worldly [$\Delta^1_1 (V_\kappa)$].
 - An inaccessible cardinal [$\Pi_1$/$\Pi^1_1 (V_\kappa)$] is worldly.
 - An $\alpha$-inaccessible cardinal [$\Pi_1$ at least for finite $\alpha$/$\Pi^1_1 (V_\kappa)$] More generally, an inaccessible cardinal of non-trivial [Carmody degree](https://arxiv.org/abs/1506.03432) [$\Pi^1_1 (V_\kappa)$] is inaccessible of every lesser degree.
 - A (boldface) $\Sigma_2$-Mahlo cardinal (equivalently $\Delta_2$-Mahlo) [$\Pi^1_1 (V_\kappa)$] is inaccessible of every Carmody degree.
 - A (boldface) $\Pi_2$-Mahlo (equivalently $\Sigma_3$-Mahlo) [$\Pi^1_1 (V_\kappa)$] is $\Sigma_2$-Mahlo. A $\Pi_{n+1}$-Mahlo cardinal (equivalently $\Sigma_{n+2}$-Mahlo) [$\Pi^1_1 (V_\kappa)$] is $\Pi_n$-Mahlo. A lightface $\Pi_{n+2}$-Mahlo cardinal [$\Pi^1_1 (V_\kappa)$] is lightface $\Pi_{n+1}$-Mahlo and boldface $\Pi_n$-Mahlo but not generally boldface $\Pi_{n+1}$-Mahlo. A boldface $\Pi_n$-Mahlo is lightface $\Pi_n$-Mahlo.
 - A cardinal that is $\Pi_n$-Mahlo for every finite $n$ is said to be $\Pi_\omega$-Mahlo [$\Pi^1_1 (V_\kappa)$].
 - A Mahlo cardinal [$\Pi_1$/$\Pi^1_1 (V_\kappa)$] is $\Pi_\omega$-Mahlo.
 - An $\alpha$-Mahlo cardinal [$\Pi_1$ for finite $\alpha$/$\Pi^1_1 (V_\kappa)$] is $\beta$-Mahlo for every $\beta \lt \alpha$, where 0-Mahlo is the same as Mahlo. A cardinal $\kappa$ that is $\alpha$-Mahlo for every $\alpha \lt \kappa^+$ is said to be greatly Mahlo [$\Pi^1_1 (V_\kappa)$] (if I understand correctly).
 - A weakly compact cardinal [$\Pi^1_2 (V_\kappa)$] is greatly Mahlo.
 - A $\Pi^n_m$-indescribable cardinal [$\Pi^1_2 (V_\kappa)$?] is $\Pi^n_k$-indescribable for all $k \lt m$ and thus $\Pi^i_k$-indescribable for every $i \lt n$ and every $k \lt \omega$, for $\Pi^{i+1}_0$-indescribable is the same as $\Pi^i_k$-indescribable for every $k \lt \omega$. Weakly compact is equivalent to $\Pi^1_1$-indescribable. A cardinal that is $\Pi^n_m$-indescribable for every $n, m \lt \omega$ is said to be totally indescribable [$\Pi^\omega_2 (V_\kappa)$]
 - A $\gamma$-strongly unfoldable cardinal $\kappa$ is $\eta$-strongly unfoldable for every $\eta$ such that $\kappa \le \eta \lt \gamma$. For finite $n$, $n$-strongly unfoldable is equivalent to $\Pi^{n+1}_1$-indescribable (this follows from Hauser's characterization of indescribable cardinals).
 - A subtle cardinal [$\Pi_1$/$\Pi^1_1 (V_\kappa)$] is greatly Mahlo but not generally weakly compact.
 - An almost ineffable cardinal [$\Pi^1_2 (V_\kappa)$] is subtle and weakly compact but not generally $\Pi^1_2$-indescribable.
 - An ineffable cardinal [$\Pi^1_3 (V_\kappa)$] is almost ineffable and $\Pi^1_2$-indescribable but not generally $\Pi^1_3$-indescribable.
 - An $n$-subtle cardinal [$\Pi_1$/$\Pi^1_1 (V_\kappa)$] is $m$-subtle for all $m \lt n$, where subtle is the same as 1-subtle or 2-subtle, depending on your convention, but not generally weakly compact.
 - An $n$-almost ineffable cardinal [$\Pi^1_2 (V_\kappa)$] is $n$-subtle and $m$-almost ineffable for all $m \lt n$, but not generally $\Pi^1_2$-indescribable.
 - An $n$-ineffable cardinal [$\Pi^1_3 (V_\kappa)$] is $n$-almost ineffable and $m$-ineffable for all $m \lt n$, but not generally $\Pi^1_3$-indescribable.
 - A cardinal that is $n$-subtle for all $n \lt \omega$ may be called totally subtle [$\Pi^1_1 (V_\kappa)$]. A totally subtle cardinal is not generally weakly compact.
 - A cardinal that is $n$-almost ineffable for all $n \lt \omega$ may be called totally almost ineffable [$\Pi^1_2 (V_\kappa)$]. A totally almost ineffable cardinal is not generally $\Pi^1_2$-indescribable.
 - A cardinal that is $n$-ineffable for all $n \lt \omega$ is called totally ineffable [$\Pi^1_3 (V_\kappa)$]. A totally ineffable cardinal is not generally $\Pi^1_3$-indescribable.
 - A completely ineffable cardinal [$\Delta^2_1 (V_\kappa)$ if I remember correctly] is totally ineffable and $\Pi^2_0$-indescribable but not generally $\Pi^2_1$-indescribable.
 - A weakly Ramsey cardinal [$\Pi^1_2 (V_\kappa)$] is totally almost ineffable but not generally $\Pi^1_2$-indescribable.
 - An $\alpha$-iterable cardinal [$\Pi^1_2 (V_\kappa)$] is $\beta$-iterable for every $\beta \lt \alpha$, where 1-iterable is the same as weakly Ramsey, but not generally $\Pi^1_2$-indescribable. The iterable hierarchy ends at $\omega_1$-iterable.
 - An [$\alpha$-club Erdős](https://mathoverflow.net/questions/425717/ramsey-theoretic-properties-of-erd%c5%91s-cardinals) cardinal (for infinite $\alpha$) [$\Pi^1_1 (V_\kappa)$] is totally subtle but not generally weakly compact. An $\alpha+1$-iterable cardinal (for countable $\alpha$) is $\lt \alpha \times \omega$-club Erdős, that is, $\alpha \times n$-club Erdős for every $n \lt \omega$ (at least I think so; my proof combines the proof of lemma 1.21 of [Friedman 1998](https://cpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2014/01/subtlecardinals-1tod0i8.pdf) with the proof of lemma 4.5 of [Gitman and Schindler](https://ivv5hpp.uni-muenster.de/u/rds/virtualLargeCardinalsEdited5.pdf)).
 - An almost Ramsey cardinal [$\Pi^1_1 (V_\kappa)$] is a beth fixed point but not generally worldly.
 - A cardinal $\kappa$ that is $\alpha$-club Erdős for all $\alpha \lt \kappa$ [$\Pi^1_1 (V_\kappa)$] is almost Ramsey but not generally weakly compact.
 - A [pre-Ramsey](https://arxiv.org/abs/1907.13540) cardinal [$\Pi^1_1 (V_\kappa)$?] is $\alpha$-club Erdős for all $\alpha \lt \kappa$ (at least I think so) but not generally weakly compact.
 - A Ramsey cardinal [$\Pi^1_2 (V_\kappa)$] is pre-Ramsey and $\omega_1$-iterable but not generally $\Pi^1_2$-indescribable.
 - An ineffably Ramsey cardinal [$\Pi^1_3 (V_\kappa)$] is Ramsey and totally ineffable but not generally $\Pi^1_3$-indescribable.
 - A $\Pi_n$-Ramsey cardinal [$\Pi^1_{n+2} (V_\kappa)$] is $\Pi_m$-Ramsey for $m \lt n$ and $\Pi^1_{n+1}$-indescribable but not generally $\Pi^1_{n+2}$-indescribable. $\Pi_0$-Ramsey is the same as Ramsey and $\Pi_1$-Ramsey is the same as ineffably Ramsey.
 - A $\Pi_\alpha$-Ramsey cardinal [$\Delta^2_1 (V_\kappa)$?] is $\Pi_\beta$-Ramsey for $\beta \lt \alpha$ but probaly not generally $\Pi^2_1$-indescribable. If $\kappa$ is $\Pi_\alpha$-Ramsey for all $\alpha \lt {(2^{\kappa})}^+$, it is said to be completely Ramsey [$\Pi^2_1 (V_\kappa)$?].
 - An almost fully Ramsey cardinal [$\Pi^2_1 (V_\kappa)$] is completely Ramsey but probaly not generally $\Pi^2_1$-indescribable.
 - A strongly Ramsey cardinal [$\Pi^1_2 (V_\kappa)$] is Ramsey but not generally $\Pi^1_2$-indescribable.
 - A super Ramsey cardinal [$\Delta^2_1 (V_\kappa)$] is strongly Ramsey and $\Pi_\omega$-Ramsey but not generally $\Pi^2_1$-indescribable and probably not generally $\Pi_{\omega+1}$-Ramsey
 - A fully Ramsey cardinal [$\Pi^2_1 (V_\kappa)$] but probaly not generally $\Pi^2_1$-indescribable.
 - A [locally measurable cardinal](https://arxiv.org/abs/2008.04019) [$\Pi^1_2 (V_\kappa)$] is strongly Ramsey but not generally $\Pi^1_2$-indescribable.
 - A measurable cardinal [$\Sigma^2_1 (V_\kappa)$] is locally measurable, fully Ramsey and $\Pi^2_1$-indescribable but not generally $\Pi^2_2$-indescribable.

I hope I'll finish this list later.