I think HoMM3 is my favorite in terms of art. 3 just had some absolutely gorgeous castle art. The best of the series in my opinion. I think 5 has the most fun unit graphics, though. They're cartoony, but I always felt that was the charm of the series, and even though they switched over to poly-based stuff instead of sprite-based starting with 4, 5 really did a good job with the unit graphics. As for music, I've pretty much liked all of them, but I think the opera selections from 2 were the best, though overall, it's a tossup amongst 3, 4 and 5 as to which has my favorite music all told. Both 3 and 5 had some good town music, but 4 had one or two towns that I just -really- liked, and a couple of pieces of map music and combat music that I also really, really liked. So, like I said. Toss-up.
Story-wise, I'd go with 4 or 5. Those were some pretty good campaigns. The ones in the first three games felt a little too disjointed for me. Still fun, but only loosely tied together. 4 also had the benefit of putting the heroes in the field, and I kinda liked the idea of having an "army" that was essentially nothing but a "special ops force" of heroes. Taking out piles of dragons with nothing but three knight heroes and three cleric heroes all at Level 20-30 was kinda fun.