The command mode is not useful to complete environments. You always have to type some letters of \begin, and invoking the completion mode will result in a huge list of environments tags. On the other hand, environments are so often used that Kile offers a special mode to complete environments. Forget the opening tag and write, for example, eq.

When you call the completion mode with Edit → Complete → Environment or keyboard shortcut Alt+Shift+Space, the opening tag is automatically added and you will see \begin{eq}. After this change, the completion list is much less cluttered.

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Now select an environment, and you will see that it is also automatically closed. Even more, if Kile recognizes it as a list environment, it will also insert a first \item tag.

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Go to Settings → Configure Kile... → Kile+Complete to configure one or more of these lists. This mode uses the same word lists as the completion mode for T_EX and L^AT_EX commands.