Air Carrier Flight Attendant reported a fume event during climb. The fumes dissipated initially but returned during final landing approach. All flight attendants experienced physiological symptoms.

Narrative

Leaving ZZZ to ZZZ1. After take off; Captain did an all-call to ask if we smell something unusual. All Fa's (Flight Attendants) said yes. Captain asked if any of us have experienced fumes in a cabin before and none of the working crew have but the jumpseater seated by 1L have and that jumpseater said it smells exactly like fumes from the plane. Captain then was checking in on us to see if he should have the plane turn around. I'm not sure how his decision making process was but we ended up continuing with the flight. Later into the flight; Captain called us (all-call) to ask each FA for what symptoms we were having. All Fa's got a headache; #4 got watery eyes as well; #3 had a weird taste on his lips. Eventually once more leveled off from take-off; the smell dissipated. However; as we were about to land; I started smelling the same fume scent again and when we were taxiing into our gate in ZZZ1; I called all FA's to see if they were smelling the fumes again. #4 said she was definitely smelling it again. I was not sure what the #3 said or the #1 said. When we got to the gate; I believe the smell was gone or not as bad anymore. After all passengers deplaned; as I was walking out of the plane; I passed by 3L 3R and all of a sudden a strong smell of the fumes again. I [could] see Maintenance was on board and left to head to operations with FA #1. We talked to Maintenance Operations Manager to see what we can do. We asked them to give us information of an urgent care address and how to go about with a report.