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What is the correct way in terms of damage to the card/have a deck well shuffled for competitive tournaments, since you can make piles; I personally consider that the best way, just one for the game.

The best way to shuffle a deck is to use multiple different methods of shuffling one after another. As the rules state pile shuffling can't be used except at the start of the game that is perhaps the best way to start your shuffle. After that the Overhand Shuffle, sliding a few cards at a time from one hand to the other so that the cards in each group stay in the same order but the group order is changed, effectively several small cuts. A Riffle Shuffle, taking two halves of the deck, bending them in the middle and letting them fall a card or two at a time onto eachother, does the best for reordering the deck, but since it involves bending the cards it also does damage to them over time, some people won't use this method because of it, and it should never be done to another person's cards for that reason. A similar effect to a riffle shuffle is the Weave or Faro Shuffle (not to be confused with mana weaving), where two halves of the deck are held with a little space between the edges of the cards for the other half's cards to slide into, this is easier with sleeves because the sleeves will create that space between the edges of the sleeves themselves. Washing the cards also does a pretty good job of randomizing, though it looks ridiculous and takes time to spread all the cards out face down and move them around randomly with your hands.

It's all a balance of the level of damage you're willing to accept to your cards, and the level of randomness needed. I will riffle my pauper deck but not my $1000+ EDH.

1) "Pile shuffling" is not actually shuffling, in that it doesn't randomize your deck. Let's say you have a deck of 60 cards and you number the cards 1-60. Then you "pile shuffle" those 60 cards into 8 piles and number those piles 1-8. Then you know for a fact that card 1 is at the bottom of pile 1, card 2 is at the bottom of pile 2, and so on, and you know that card 9 is second from the bottom of pile 1, then card 17, 25, 33, 41, 49, and 57. Since you deterministically know the order of the cards in each pile, this method is not considered randomization, and is therefore not "shuffling". This is why judges (and some players, thankfully) have begun to call the method "pile counting"; it is an efficient method of counting the cards in your deck to make sure you haven't lost any, but that's all.

2) As a result of point 1), you are not allowed to "pile shuffle" your deck in a tournament for the purpose of randomization. You are allowed to do it for the purpose of counting, but you still need to shuffle thoroughly afterwards.

3) As for an actual answer to your question, the answer is "use sleeves". If you use sleeves, damaging your card through standard mash shuffling is very uncommon, and mash shuffling is the most efficient way to shuffle, in that you are shuffling a random number of cards from the deck into random positions in the remainder of the deck. Since you may not fully randomize the bottom few cards of the deck using a mash shuffle, you should also use other shuffling methods such as overhand shuffle. Personally, I haven't found bridge (riffle) shuffles to damage my cards all that much, as long as you don't angle your bridge too high and bend the cards too much.

4) As an addendum to "use sleeves", sleeves can sometimes rip during a mash shuffle and the sharp edge of the sleeve could nick the card inside of the broken sleeve. In which case, you can use perfect-fit sleeves ("inner sleeves") inside of your normal sleeves. Then when the outer sleeve would nick your card, instead it nicks your inner sleeve and keeps your card safe. I've never damaged a card using a mash shuffle when my deck was "double-sleeved" (using both regular sleeves and inner sleeves).

How to shuffle (theoretically)

It takes 7 riffle shuffles to randomize a deck of (52) cards. If you're playing a 60 card format 7 riffle/mash shuffles are sufficient. If you're playing commander which uses 100-card decks, you'll need to riffle shuffle 9 times to be randomized enough. Shuffling more does make the deck more random, but 7 (or 9) iterations is enough in practice.

Pile "shuffling" does not randomize cards and is a waste of time. Just perform more riffle shuffles instead.

How to shuffle (physically)

Riffle shuffles are easy enough to do with practice. You don't actually need to "bridge" the card as long as the cards are interleaving frequently and non-perfectly. I often shuffle at the corners/sides and then just push the cards together.

As described in other answers, sleeved cards "mash" together easily to effectively replicate the mechanics of a riffle shuffle. Cut the deck in half and push one half into the other block of cards. They will naturally interleave, imitating a riffle shuffle.

If you play commander and need to shuffle 100-card behemoths, shuffling is harder. Unsleeved (*gasp*) commander decks can still be riffle shuffled fairly easily. And if you have large hands you might be able to mash shuffle a sleeved 100-card deck. Otherwise, you'll have to get creative. My best advice is to split the deck into two 50-card piles. Then when you riffle/mash shuffle, shuffle together the top halves of each of the piles and repeat with the remaining bottoms. Repeat 8 more times.