An acrostic poem should tell some sort of story. Even at the most basic level an acrostic poem is capable of saying more than simply listing words that start with the correct letter, however, children especially often fall into that trap. Poets should choose a type of story they want to tell with their poem and then find the words to communicate that story. Otherwise, the poem will come off as extremely simplistic.
Make sure there is a connection between the word that forms the foundation of the acrostic and the lines that you write that start from the letters of that word. This is why it is difficult to write an acrostic poem well from an ordinary or uninteresting word, such as "the". Obviously, "cat" is a simple three letter word that would provide much more inspiration for a poet than the un-evocative article, "the."
The simple form of an acrostic contains only a single word on each line of the poem. As you get comfortable with that type of poetry, try to express more complex thoughts in each line. Instead of a single word, write an entire line of poetry where only the first word starts with the designated letter from the original word of the acrostic. Write freeform, or create a more interesting challenge and assign yourself a rhyme scheme that your lines need to adhere to.
If you struggle with writing acrostic poetry, read some of the best examples of the form to see what is possible. Acrostics don't need to be simple or elementary, and can actually be quite beautiful. Look at Edgar Allen Poe's "An Acrostic" for one example. Another example of a very effective acrostic by a well known writer is "A Boat Beneath a Sunny Sky" by Lewis Carroll. Here, the inspiration for Carroll's most famous creation, Alice, has her name used as the base for the poem.