How do I list every file in a directory except those with specified extensions?

Assuming one has GNU ls, this is possibly the simplest way:

ls -I "*.txt" -I "*.pdf"

If you want to iterate across all the subdirectories:

ls -I "*.txt" -I "*.pdf" -R

Find supports -o

find . ! '(' -name '*.txt' -o -name '*.pdf' ')'

You need the parenthesis to make the precedence right. Find does a lot of stuff; I suggest reading through its manpage.

You can also do an or in grep (but really, you should not parse the output of ls)

ls | grep -Ev '\.(txt|pdf)$' | column

With bash extended globbing (turn on with shopt -s extglob), the glob !(*.txt|*.pdf) should work. You can pass this glob directly to any command that takes file arguments, including but not limited to ls.