What do I need to do to understand T. S. Eliot's poetry well, as with "The Waste Land"?
To understand poems by T. S. Eliot, such as "The Waste Land" well, you need to be familiar with their cultural and literary context. Eliot is a highly allusive poet, who constantly refers to other works of literature, art, history, philosophy and religion. Recognizing those cultural referents will make it easier to understand his poetry.
Next, Eliot wrote a great many critical works about poetry. The more you read of Eliot's criticism, the better you will understand the underlying philosophical and critical ideas that inform the poems. A particularly important essay, that explains, inter alia, how and why he uses allusions, is "Tradition and the Individual Talent."
Another important concept for Eliot as a poet is that of the disassociation of the sensibility and the fragmentation of culture, making the monumental edifices of earlier art and literature impossible, a phenomenon he references with the famous line near the end of "the Waste Land":
These fragments I have shored against my ruins