If I score a Battlefield on my turn and then move everything away from it and retake it on the same turn, do I get Conquer effects (Rek'Sai for example)?

No, you do not trigger conquer effects (such as those on a card like Rek'Sai) if you retake a battlefield you have already scored earlier in the same turn.

According to the established rulings in the game:

• You can only score each battlefield once per turn. (Rule 447)
• Conquering is a subtype of scoring. (Rule 446.1)
• Conquer effects only trigger when you successfully conquer a battlefield. (Rule 376.4.b.2.a)

Because you have already scored the battlefield once that turn (either by holding it or by a previous conquer), you cannot score it again. Since you cannot score it again, you cannot perform the "Conquer" action, and therefore any "When you conquer" effects will not trigger. This remains true even if you leave the battlefield and retake it later in the same turn.