to get better
(=recover)
[sick person] se remettre, se rétablir
→ The doctors said there wasn't much hope of me getting better.
→ Don't worry, you'll soon get better.
I hope you get better soon. J'espère que tu vas vite te remettre.
(=improve)
[person, situation] s'améliorer
→ The match started off pretty badly, but it got better.
→ She's a good teacher, and getting better all the time.
I hope the weather gets better soon. J'espère que le temps va s'améliorer bientôt.
My French is getting better. Mon français s'améliore.
to be better
[sick person, patient] aller mieux
→ I'm much better, thank you.