There are quite a few characters I didn't like at the beginning of their shows and grew to like over the course of the show, as they developed.
In DS9 - Bashir (I found him annoying and badly-acted in season 1, started liking him around season 4), Jadzia Dax (I found her dull in season 1, then I found her a bit better, then realized I really liked her around season 4)
I had a peculiar experience with ENT, when I first tried to watch the pilot, I hated it, couldn't even watch the episodes all the way through, and I disliked most of the characters - I hated Archer and Trip, didn't like Reed much either, and had an impression that Phlox was going to be an annoying wacky-alien-comic-relief character like Neelix. Then I decided to give it another try a few months later, watched it again and liked it this time. The first time I thought the (male) human characters were ignorant gung-ho jerks, the second time I watched it I found it refreshing that they weren't like the 24th century Trek people. Maybe it was because I had watched VOY in the meantime. I soon realized that Phlox is actually awesome and nothing like Neelix, and most of the characters really grew on me (apart from Travis since he was hardly a character). Instead of an annoying moron, I found to be Archer an endearing moron

and he got much better and less stupid over time
Is there a category "a character you really hated, and then grew to dislike less"? I thought Neelix was really horrible in the early seasons of VOY, the worst character in Trek ever - but he got tolerable since Kes left. He was always at his most annoying around her. I still think he's a badly conceived character whose role on the ship makes no sense, but in the later seasons of VOY I don't get the urge to curse whenever I see him.
As for characters I liked at first but not now... Well, the first time I watched TNG (it was season 1), I was about 12 or 13, so I liked Riker because I found him hot (though I found his posturing a bit silly even back then) and Troi because I found her gorgeous, and I shipped them. (What do you expect from a 13-year old girl?) Then when I grew up and watched it again, I was indifferent to Riker and I realized just how lame Troi is as a character, especially in early seasons. Though they both get better later on, especially Riker who gets more likable - and better-acted.
Wesley.
I know, I know. However, I was about half Wesley's age when TNG premiered, and I looked up to him as relatable. Not the MOST relatable, that would go to Mr. Data, but relatable nonetheless.
Funny, even as a 12/13-year old, I was completely indifferent to Wesley. I didn't hate him, but I just hardly noticed him. He was just... there.