On Linux I use Fcitx and Ctrl+Shift+Alt+U to open the Unicode character picker, while on Windows I have my fork of a character picker discovered a while ago (https://github.com/flobosg/character-picker) bound to the same shortcut.
Most linux desktops are different and more complicated but here is how I set mine up on openbsd.
pick a key that is handy but you don't use much, for me this was the context menu key, on many european keyboards there is an altgr key set aside for this exact use case.
in ~/.xsession add a line like "xmodmap -e 'keysym Menu = Multi_key'"
create a file ~/.XCompose to setup your own compose sequences. mine looks like
#load system compose files
include "%L"
<Multi_key> <w> <e> <b> : "\xf0\x9f\x95\xb8" # spiderweb