In my opinion, HOMM3 is the best game to start out on. HOMM2 is another good option, but I think many will agree with me when I say to start with 3. I will leave out #5 and #6, as those are games you should only play if you've played previous installments. They are DEFINITELY not good starting points, as true appreciation for the series will come from paying homage to the games that came before it.
Here's my breakdown of the series:
#1 (The Original) - Unless you played this game when it first came out, thereby introducing you to the series, I would not recommend starting with this one. As is to be expected, it has become outdated, and the gameplay is slow and tiring comparatively speaking.
#2 (The Middle Child) - This is still a solid piece to the series. It has a great art style and it's easy to pick up. The character models improved significantly from its predecessor. Gameplay is solid and it's a game I often find myself coming back to. It is often overlooked, due to the incessant love for #3, and the bad-taste-in-your-mouth vibes for #4. I often see it as the child that gets the least attention.
#3 (The Fan Favorite) - This game is often the fan favorite, and for good reason! Graphics are great and easy on the eyes (which is a good thing since this game is so addicting), gameplay is fantastic, music is memorable, and the story (campaign) is great. I have every installment in the HOMM series, and this is by far my favorite and the game I find myself playing most often. If you are looking to get into the series, this is a great game to learn from, since it is the measuring stick that many fans use to grade sequels.
#4 (The Black Sheep) - This game gets a lot of hate for deviating too much from #3. However, if you include the expansions, this is a pretty solid game. It introduces a lot more strategy than its predecessor, which is evident by the way it handles builds (no creature upgrades). Don't write this one off, though. If you like #3, #4 will grow on you.
All in all, I would start with either #2 or #3. Those are the games I was introduced into the series with, and I believe that they will give you a really good taste of what the series has to offer.