Familiarize yourself with other poems of the style. This is perhaps the best way to learn the conventions of the medium and also to get some ideas for your own poem. Pay attention to what's working in other people's poems and what is not, as this will help inform how you write your own. Academic databases, such as the palindrome poem examples given online through Michigan State University's poetry program are good sources for finding reference poems.
Decide which format of palindrome poem you'd like to create, and if you want every letter of your poem to be the same forward as backward, or whether every word is enough. Just focusing on every word is less restrictive, but also perhaps less rewarding in the end. You could also use a combination of every letter for some lines and every word for others, as Demetri Martin did in his palindrome poem, "Dammit, I'm mad."
Figure out a subject, topic, or theme for your poem, and start writing. Your topic can help inform your style and method of writing. For example, a humorous topic might work if the poem is nonsensical or whimsical, but a very serious topic might need to stick to writing conventions.
Use online or print resources if you get stuck. A dictionary can help ensure that you're spelling everything correctly, and a thesaurus can allow you to say something in a new way that might work better in your work.
Read over the poem multiple times once you've finished and check to make sure everything does actually work as a palindrome. Misspelled or omitted words can easily drain the impact from your poem and undermine your work.
Share your palindrome poem with the world. Facebook, blogs and even mediums like YouTube can help you get your poem out there and can showcase your creation to any poet interested in this tricky style.