cranks over slowly then goes faster

I own a 2003 Hyundai Accent .Recently purchased it .It cranks over slowly at first and then faster after a second. Battery tested at 83% so replaced it with a new one from Walmart and the new one was almost dead ,Had to charge it, The car still cranks over, I am thinking the starter motor or maybe the battery could the timing have anything to do with the slow cranking? I had a new timing belt put on at the Hyundai dealer on Wednesday.