How does the Steenrod algebra act on $\mathrm{H}^\bullet(p^{1+2}_+, \mathbb{F}_p)$?

If $P$ is the group of order $p^3$ and exponent $p$, its mod $p$ cohomology ring is known to have its depth = its Krull dimension = rank of a maximal elementary abelian subgroup = 2. A theorem of Jon Carlson then implies that the product of restriction maps $$ H^*(P;\mathbb F_p) \rightarrow \prod_E H^*(E;\mathbb F_p)$$ is monic, where the product is over (conjugacy classes of) subgroups $E \simeq \mathbb Z/p \times \mathbb Z/p$.

Thus the ring you care about, viewed as an algebra equipped with Steenrod operations (I would call this an unstable $A_p$--algebra) embeds in a known unstable algebra. Thus if you know how algebra generators of $H^*(P;\mathbb F_p)$ restrict to the various $H^*(E;\mathbb F_p)$'s, it shouldn't be hard to calculate Steenrod operations.

For example, David Green and Simon King's group cohomology website tells you cohomology ring generators and relations, and these restrictions for the group of order 27. (See https://users.fmi.uni-jena.de/cohomology/27web/27gp3.html) I'll let you take it from here.

[By the way, you began your question with a remark that there isn't so much to the cohomology ring $H^*(C_p;\mathbb F_p)$ as a module over the Steenrod algebra. Yes, it is trivial to compute, but it is a deep theorem, with huge unexpected consequences, that this is a injective object in the category of unstable $A_p$--modules. See the book by Lionel Schwartz on the Sullivan conjecture for more detail.]

Edit the next day: Thanks to Leason for pointing out that my map doesn't detect for $p>3$. To get the detection result, one needs that the depth = Krull dimension, and this turns out to only happen in the one case that I looked up carefully: $p=3$. So in the other cases, the depth will be 1 = rank of the center, and one needs a more general detection theorem, pioneered by Henn-Lannes-Schwartz in the mid 1990's, and then explored by me in various papers about a decade ago. (Totaro later wrote about this in his cohomology book: this is the result mentioned by Heard.) In the case in hand, the range of the detection map for $H^*(P;\mathbb F_p)$ will need one more term in the product: Let $C < P$ be the center: a group of order $p$. The multiplication homomorphism $C \times P \rightarrow P$ induces a map of unstable algebras $$ H^*(P;\mathbb F_p) \rightarrow H^*(C;\mathbb F_p) \otimes H^{*\leq 2p}(P;\mathbb F_p)$$ where the last term means truncate above degree $2p$. That the number $2p$ works to detect all remaining nilpotence is an application of my general result: it can be determined by understanding the restriction to the center.

At any rate, for that original group of order 27 and exponent 3, this isn't needed. (The group of order 27 and exponent 9 will also need that extra factor in the detection map range.)

I think the answer you want can be found in the following article:

AUTHOR = {Leary, I. J.},
 TITLE = {The mod-{$p$} cohomology rings of some {$p$}-groups},
JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical
          Society},
VOLUME = {112},
  YEAR = {1992},
NUMBER = {1},
 PAGES = {63--75},
  ISSN = {0305-0041},