Obviously, if you exclude free counters, there is nothing you can do against first turn discard when you are on the draw. There are several answers to targeted discard after that, and before the match even starts:
Play a deck without cards whose loss would cripple your strategy/combo
Sligh comes to mind, or White Weeny, or Stompy. Anything but Combo, basically, preferably Aggro. If there are no key cards in your deck, you can't lose them to discard.
Use a deck that benefits from discard
Nowadays, there are many cards and whole mechanics that benefit from you having few cards in hand or many in your graveyard. Cursed Scroll, Ensnaring Bridge, Crucible of Worlds, Tarmogoyf, Hellbent, and Threshold are examples.
Draw more cards than he can make you discard
All card advantage strategies apply.
Tutor for key cards
Nearly all discard, and AFAIK all targeted discard has sorcery speed. Use instant tutor cards like Mystical Tutor to get you the cards you need exactly when you need them, without an opening for your opponent. That increases your flexibility and allows you to carry 1 each of a wide range of strong cards.
Use targeted recycle to get the discarded card back
Use cards like Regrowth or Noxious Revival to simply return the card you had to discard. Either you get the card back immediately, or, if you have to put it on top of your library, play the recycler at the end of the opponent's turn. If enough cards are in your graveyard, this even acts as a kind of tutor, since you may get to choose from a variety of cards. Recyclers are not wasted, either, when your opponent doesn't play discard, so they can be easily integrated into the main deck.