A Good Shuffle
There's a few very good reasons to get a good shuffle. The first and probably the most important is so that your opponent can trust a good game was played. If you do happen to win that game, your opponent can't possibly accuse you of cheating through a stacked deck. I don't know about the rest of you, but I play to win. But winning is only important when the opponent realizes that you won. Winning by cheating is pointless and stupid. It proves nothing except for that fact that you probably couldn't win without cheating.
The second big reason to get a good shuffle is for deck uniformity. By that, I mean that your deck is actually randomly shuffled. If you have four Electabuzzes, a random card deck will yield a decent percentage of you getting the card on the first draw.
Well, why wouldn't that happen anyway? In many games, the first ten cards you discard are mostly Trainers. Occasionally you will discard twenty cards including Energy Cards and Trainers before your first Pokemon gets knocked out. Well, if you don't shuffle well, that large clump of non-Pokemon may appear on top and you may lose a few mulligans the opening round, which is enough to possible end the game before it started.
So one of the things I do when discarding is to actually "pre-shuffle". If my first Pokemon card goes to the discard pile, I'll actually insert it into the initial non-Pokemon Card area. I don't believe this is against the rules, so there's nothing wrong with that. By doing this, your cards will actually be semi-shuffled before the game ends. It really helps in shuffling the cards.
If you have 14 or more Basic Pokemon in your deck and you get frequent mulligans, a poor shuffle is probably the reason. Make sure to either shuffle well, or separate Basic Pokemon when you are discarding them.