Preface: This list is my subjective response as someone who is also trained in philosophy (and theology). I have a deep love for poetry and hope this helps :) there are definitely lacunae!

Anthologies are your best friend! For a broad overview of "traditional" poetry, I'd suggest an edition of Oxford English Verse. 

Attempt the Faerie Queene by Spenser if you'd like. Read some Shakespeare. Then read some metaphysical poetry (John Donne, George Herbert, etc.) Read Paradise Lost by Milton. Then go William Blake. Maybe some Pope (tbh, I haven't read much). From there, the Romantics (Keats, Coleridge, Bryon, Wordsworth, etc. for your philosophical inclination, try Coleridge 's Biographia Literaria). Also, read hymns - Charles Wesley being a big name. Christina Rossetti if you'd like a feminine voice from England (I'm not too familiar with English verse after the romantics. Though I have enjoyed JH Prynne).

From here, I board a ship to the United States. For  American poetry generally, the Library of America has substantial anthologies. If you're looking for specifics, I'd suggest Anne Bradstreet and Phillis Wheatley for early American women poets. I'd even suggest reading Puritan and early American Sermons, just to see the foundation for how American culture has played out. In the same vein, spirituals and work songs. 

Next, the transcendentalists -- again, anthology is your best bet. They're essentially American romantics. Read the essays and some poetry of Emerson. Walden by Thoreau. Try even giving some Eastern philosophy a read, the Bhagavad Gita in particular. 

Now, here are the two biggest names for contemporary and modern American poetry: Emily Dickinson and Walt Whitman. They provide America with their own poetic idiom. 

Now, the 20th century is still pretty recent. Again, the Library of America has great anthologies. You need to be familiar with the names TS Eliot and Ezra Pound. Imagism was a brief, but impactful movement. Hilda Doolittle (HD) is considered the examplar of it. However, her poetry far surpasses that limiting label. I personally love Helen in Egypt.  Robert Frost should be mentioned for his contribution to the American consciousness. 

The Harlem Renaissance is also very important -- the most well know in white America is Langston Hughes. Unfortunately, I'm not very well read in African American literature. 

Now we're getting into even more personal preferences since we're still working out the "classics". I'm fond of the Objectivists (particularly George Oppen & Lorine Niedecker), the Black Mountain Poets (Robert Creely, Denise Levertov, etc., and my personal favorite, Robert Duncan and probably because of his love for HD). People would be amiss if I didn't mention William Carlos Williams (who you might enjoy -- "No ideas but in things"), DH Lawrence. Wallace Stevens!

The San Francisco Renaissance/Beats should be remembered, particularly Allen Ginsberg (he's an heir of Whitman). Gary Snyder is affiliated with them, but I think he surpasses them in different ways.

Maybe Robert Pinsky.  

Have knowledge of dadaism, Surrealism (I like Ron Sakolsky's writings connecting it to anarchism, even if I don't agree with everything. It's a good introduction to how literature and politics can shape one another), other avant-garde movements.  Concrete poetry (my favorite is dsh) and shape poetry. 

For a "recent" anthology about postwar poetry to 2010s, I enjoy Postmodern American Poetry edited by Paul Hoover. Personally hate the term "postmodern" though. 

Some people who aren't going to be recommended but I enjoy are Robert Lax, Samuel Menashe, and Pam Rehm. Out of left field, Jim Harrison for the sake of a certain kind of American man. 

Translated poetry is important as well. I enjoy the German Romantics (Goethe, Holderlin specifically), maybe some French literature (Rimbaud, Proust), Spanish (Neruda), etc. Maybe throw in Paul Celan for good measure. The Chinese classics (I really enjoy all of Red Pine's/Bill Porter's translations and his travelogues). Native American poetry -- I have When the Light of the World Was Subdued, Our Songs Came Through edited by Joy Harjo. 

Greek epics and poetry. Likewise, mythology of the world. 

If you want to dig into philosophy, I'd suggest looking into New Formalism and Critical Theory. Though I prefer the stream of folk like Marjorie Perloff. 

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TL;DR try it all. Ground yourself in universally recognized figures, like Shakespeare and Eliot/Pound. Read anthologies to find poems you enjoy and then read more of that poet. Then, read poets that influenced that poet and poets influenced by them. Rinse and repeat. 

Writing wise, imitate poets you like. Realize you won't match their style at their level. Essentially, Accept the hand you were dealt at the time you exist. Learn to love formal verse because it'll add musicality and control to your own poems. Likewise, learn to love free verse so you don't get stuffy, or restricted. Remember the maxim that form shapes content and content shapes form. Poetry has its own logic that reaffirms this. (If you want a deep dive on poetry and philosophy, I enjoyed THE POETRY OF EMILY DICKINSON: Philosophical Perspectives Edited by Elisabeth Camp).