Should I play Diablo 1 and / or Diablo 2 before playing Diablo 3?

Solution 1:

No.

While the series' lore continues, the series was never known for its story. If you want to experience the series, you will most likely need to play alone or with friends as D2 online players rush through the game. While you can let experienced players run ahead, they will open and close quests for you before you can arrive, and spend little time looking at new items and listening to dialogue and cutscenes (which, undoubtedly, is why you're playing). Also the spam bots are awful. The only benefit to you now, pre-Diablo III, is whatever you happen to take away from that rushed experience. (Even the Easter eggs are obscure and most likely won't be things you remember from a normal playthrough.) And even after all this, D2 is a very long journey to undertake alone.

The game mechanics are also different between each game—many of the dog-eat-dog elements of D2 (stealing loot, final boss rushes, power leveling) are downplayed or simply removed in D3. Unless you would play the games anyway, you will benefit more from the time invested by watching a few D1/D2 gameplay videos for nostalgia and this lore video series instead:

Solution 2:

No.

If you are new to the universe of diablo, this is one of the best introductions you can find:

It explains the diablo universe and what happenened in the two previous diablo titles without spoiling. It also goes over some games mechanics and development.