The trick to winning in a game of Trippin is to run your opponent low on cards or so that he only has fewer options in his or her hand. Since there is usually no draw step, any card that is cast but does not resolve, means one less card for your opponent. Remove the targets of his spells so that the cantrip fizzles and no cards are drawn whenever you can. Eventually, your opponent will be unable to stop your creatures.
Knowing what cards, and how many copies of each card are in the deck helps when anticipating what cards might be played against you and how to fizzle them. You can sleeve each card in a sleeve of its own color so players can better guess what cards their opponent holds in hand. This also facilitates the Long Trip optional rule.
There is seldom a reason (but not never a reason) not to play a spell as soon as you can since it will be replaced as long as it resolves. Play cards judiciously. Deal an extra point of damage to your opponent when you can if the damage does not cost you a card and will not currently aid you with the creature battle. That being said, there are a number of cards that are powerful enough in the right situation to make it worth holding on to in anticipation of that situation arising. Nearly any spell that can remove a target or make a target illegal should be saved for those moments where that is a possibility. Other cards worth holding on to for key moments include, Abeyance, Aphotic Wisps, Arcane Denial, Burnout, Cease-Fire, Dark Dabbling, Equal Treatment, Remand, and Shatter Arc.
Arcane Denial and Remand, though they appear to be counterspells, are almost always better used on ones own spells in a game of Trippin, especially when that spell would be countered or fizzle upon resolution otherwise. These are one of the few ways to add extra cards to your hand. In a pinch, Arcane Denial can be used to counter an opponents spell, but Remand on an opponents spell can, at best, only reorient that spell to the top of the stack or wait to start a new stack, since the countered spell can just be played again without mana.