Inputting Unicode in X Windows is easy. Just type:

  setxkbmap -option compose:ralt

Now, right Alt (AltGr) is your compose key.

Then, for example:

  à is Compose + ` + a
  ö is Compose + " + o
  ß is Compose + s + s
  ç is Compose + , + c
  € is Compose + = + e

In Emacs, the easiest way to input Unicode directly is to set read-quoted-char-radix to 16 in your .emacs. Then use C-q (typically bound to quoted-insert) to input a glyph directly via hex code point.

  ∀ is C-q + 2200
  λ is C-q + 3bb
  ∃ is C-q + 2203
  若 is C-q + 82e5
  世 is C-q + 4e16

An easier way is to dial up a character map program like gucharmap and copy and paste.