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I used to write poems. I was in 4th grade, so I just wrote for fun. They didn't have any figurative language or symbols. My only goal was to make them rhyme. Here is an example of one of mine:

The topic was risky

The judge wasn’t blank, but picky.

My stomach was getting really funky,

My mind was full of question,

And when my belly was acting like a monkey,

I was full of tension.

Then it all struck me,

Just like an earthquake,

I wasn’t making a single mistake.

The idea was as easy to find as in my pocket,

The idea was obviously Sonny Crockett!

Now that we have read Shakespeare and Edgar Allan Poe, I know there is a lot more to writing a poem. For it to be a good poem.

So, I decided to write poems again, but it is really hard for me to think of a symbol, and put it into my writing without revealing too much, so I can let the reader infer, but revealing enough.

My question is: How can I give my poems more depth and symbolism?

There is a character-building strategy that uses Theory of Mind where an important character is described only through the perception of another less-important character. It forces you to imagine someone through the eyes of another, limited by the secondary character's vocabulary and experience. Symbolism is one way to pass clues to the reader past the less-informed narrator, but it will also lead to a better narrative voice which can be disassociated from the subject.

A family holiday seen through the eyes of a child will be different than a grandparent and very different from a teenager. Poetry isn't just wordplay as you have realized, poetry can be uniquely non-narrative in the way it talks about subjects. You don't need to tell a "story" or even relate events in a naturalistic way, it can be loaded (or biased) heavily in one particular style or POV to the point where the "truth" is completely obscured, or at least less important than the impressions.

Theory of Mind is also a great way to explore an unreliable narrator, where the author still needs to convey an idea of what is true even when the narrator does not understand it.

The reason your 4th grade poems didn't have any depth is you weren't putting any in there. As you said, they were just fun bits of doggerel that rhymed. If you want deeper poetry, choose a deeper topic: Heartbreak, spirituality, acceptance, isolation, depression, love, hope, beauty, nostalgia, friendship, and so forth. In my experience, you'll typically do better in poetry with an emotion-driven topic than an intellectual or conceptual one.

Next marry it to some specific sensory imagery or experiences --the feel of the weather outside the last time you saw someone, or the smell of childhood. Then decide what structure you want --something with strict rhythm and rhyme? Or free verse?

The artistry comes in terms of how well and how naturally you bring those very different things together (topic/theme, imagery, formal structure).

In my own very brief experimentation with poetry, I always found it helpful to start with the image, the symbolism, as it were. So I wouldn't be "giving my poem more symbolism" - I'd start with the picture in my mind, and write the poem around that. In that fashion, I could replace words with synonyms, shift words around, scrape everything and start from scratch with a different rhyming scheme, and the core idea of the poem would remain the same.

So, for example, in this poem (written in highschool), I started with the image of "war orphan himself going to war - phoenix, dying so his son can be born":

Like a phoenix, I was born from the ashes.

When the fires of battle have died,

In the air rang my life's first cry.

I have never known my father -

The phoenix is orphaned from birth,

But I've been raised like all others,

Learning both of sorrow and mirth.

I have been like all other teenagers,

Finding true love's kisses' first joy.

I've got work, I am earning my wages,

And my wife has begotten a boy.

Lo! Of battle again rings the cry.

In the fires of battle I die,

But my son will be born from the ashes.

It's not a good poem, but because it is not good, it is easy to see in it how everything is built on the image at the core. (In Shakespeare's work, everything fits together perfectly and seamlessly, so it's harder to pick it apart and say "this came first".)

It's not an approach that would always work. For example, it might be that you are trying to tell a story in verse. In such a case, the story, not a particular picture, is at the heart of what you're writing, and everything else has to embellish that. But it is one approach, that you might find helpful, if only for practice.

What is depth?

Drag the bucket up;

Sloshing water dripping darkly down,

Back into the shadows of the well.

Slake your thirst;

Gulp the water down -

Chilly and refreshing, drawn right up

From the depths beneath - an endless well.

If you say

Only just one thing,

Like a stone, that can't be anything

But a stone, your words will have no depth.

Tell a tale -

What else does it mean?

If there is no answer, then it means

Only what you said - it has no depth.

Metaphor,

Archetypal shapes

Lurking back behind the front-most shape -

Will that give a poem deeper depth?

What is depth?

Drag the bucket up.

Water which you did not draw before

From the deep and shadowed selfsame well.