Speak English Like a Native

When a person is sad because they are rejected by someone with whom they consciously/subconsciously wanted to procreate with/enjoy the relational and pleasurable tools of procreation, we call it heartbreak.

Breaking up
This territory usually has very culturally-distinct metaphors. For example, in Chinese when people break up, it translates to "they blew." In English, "to blow" someone is to give oral pleasure to a phallus. (Contrarily, to blow someone off is to ignore them or skip plans with them.)

The most casual way to say this is to "dump someone." It's rude to say it to the person you are breaking up with, because it implies that they are like disposable waste. Breaking up or splitting up are more polite phrases.

What to say

There's a cliche in English that the one doing the breaking up always says "It's not you; it's me." Avoid that phrase, no matter how true it may feel.

If nuanced English isn't your bag, I would recommend saying "This isn't working out." That acknowledges the multiplicity of vibes that must work out for a relationship to exist.

Marriage
When married people break up but don't get a divorce, they usually say they are "spending time apart" or "taking a break" or having "an experimental separation." These are mature phrases that acknowledge the trickiness of the situation, and confuse their children.

If you're dumped

If you're angry .. Say "Go to hell!" Commanding someone to go to the Christian land of afterlife torture and punishment is an affective way of expressing how immoral/worthless you think they are.
If you're also feeling like your relationship wasn't working out ... Say "I was sensing the same things. The feeling is mutual." Then you're not quite dumped anymore.

To learn more about heartbreak, examine closely these treasured English-language songs:

"Love Hurts" - Gram Parsons with Emmylou Harris
"Jealous Guy" - John Lennon
"I Just Don't Know What to Do with Myself" - Dusty Springfield or The White Stripes
"No Scrubs" by TLC
"Ain't No Sunshine"
"Ex-Girlfriend" - No Doubt
"The Cure" - Pictures of You
"Maps" - Yeah Yeah Yeahs
"Love Will Tear Us Apart" - Joy Division
and of course, "How Can You Mend a Broken Heart" - Al Green