To delete some parts of a document you can of course select them, and then use the Delete key. Kate also offers the command Ctrl+K which deletes the whole line. But Kile offers a faster way with its own delete commands. Under EditDelete you will find the following commands to delete text.

Edit->Delete items
Environment (inside) (Ctrl+Alt+T,E)

Delete an environment without the surrounding tags.

Environment (outside) (Ctrl+Alt+T,F)

Delete an environment including the surrounding tags.

TeX Group (inside) (Ctrl+Alt+T,T)

Delete a TEX group inside the surrounding braces.

TeX Group (outside) (Ctrl+Alt+T,U)

Delete a TEX group including the surrounding braces.

Math Group (Ctrl+Alt+T,M)

Delete the current math group including the math commands.

Paragraph (Ctrl+Alt+T,P)

Delete a whole paragraph. Look at the SelectParagraph command, how a paragraph is defined in Kile.

To End of Line (Ctrl+Alt+T,I)

Delete the text from the current cursor position to the end of the line.

TeX Word (Ctrl+Alt+T,W)

Delete the word or LATEX command under the current cursor position.