I'm wondering, which verb is the most common one and the best one with the phrase "to a whole new level"?
- This discovery will (lead,bring,take,raise) evolution to a whole new level.
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You're just looking for a verb of some kind - the one you use depends on the context.
Any of the words you suggest could work along with a bunch that you haven't listed, like kick, shove, move, throw, etc.
The phrase, "to a whole new level," assumes that a thing has been at one point, but something is changing and causing it to move to a new point, or new level.
For example: I was on a special diet, but my weight hadn't changed for a long time until I started exercising. Exercising pushed my weight loss to a whole new level.
Exercising moved my weight loss to a whole new level.
Either one sounds fine. It all depends on what kind of emphasis you want to give.