I think giving up, moving on, letting go, etc. are all different things but it kind of depends on what each one means to you personally.
I focus more on acceptance, allowing, and appreciation. Accepting that things are what they are and allowing it the space to be - knowing that change is inevitable - and allowing it to change. Accepting my own feelings about it, allowing myself to feel however I feel and allowing those feelings to change.
Accepting that other people make their own choices and accepting the choices that I've made. And allowing myself to keep making new choices and to choose differently when I feel like it.
And finding something to appreciate - rather than focusing on what seems bad or wrong about the situation - this can be really hard to do sometimes, but it really changes my perspective when I can do it. It can be anything - something big, something small, something unrelated to what is bothering me, or even appreciating that something upsetting came into my life for a reason and give me an opportunity to grow or learn something about myself.
Appreciation can make you feel better, but more so it shifts the energy and takes your attention off of what is "wrong" so it has the space to change and transform.
The times when it feels hard to appreciate anything, are usually when I'm not accepting things as they are, and/or not allowing my own natural feelings about the situation.
Sometimes when people talk about giving up or moving on - it sounds like it is coming from a place of NOT accepting how they really feel and trying to push themselves to get over something. I think when a person is ready to move on - whatever that means to them in their situation - they just DO it. It's not a big decision, it just happens naturally.