Yes, yes you absolutely should. The Half-Life series is one of the greatest gaming franchises ever created, and you will love every second that you spend with it. Personally I don't understand what people are talking about when they say that the games haven't aged well: Half-Life 2 still looks and plays better than most games made nowadays. Heck, Half-Life 1 was made back in the 90s and it still holds up pretty well, so long as you can get past the graphics. So yes, you should definitely play them.
As for where you should start, I would say that you should probably start with Half-Life 1. You'll need to ignore the dated graphics (or play the new mod, Black Mesa), but I feel that playing it is vital to fully appreciating the story of the series. You can probably get away with starting at Half-Life 2, but I would still try the first one if you can.