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

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{al}. After this change, the completion list is much less cluttered.

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.

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.