Aw, shucks. Can't you give us a hard one to work on!
Just kidding, but this is an extremely common valve...a Moen legend pull-out. The handle comes off by removing the round cap ( pry it with a knife) then take out the screw. A sleeve will then pull out, to expose the tip of a bronze retaining clip. After turning off main water supply, you can pull out that clip, then pull out the cartridge. Sometimes a special puller tool is needed if it is stuck.
The handle and cartridge are readily available and inexpensive. The square escutcheon may be harder to find, but a round plate can be used.
Your handle is an aftermarket replacement, which is why it doesn't say "Moen" on it.