In order to unlock all trials, you must first complete the main levels of that portion of the game (ie complete the Peggle Institute to unlock all of the Peggle Institute Trials)
Long and Long - Use Bjorn's Super Guide to get 2 long shots, which you can get with one ball if you aim from the top left peg, to the middle right peg properly, as it will bounce to the bottom left peg and then possibly the bottom right peg (3 total bounces).
The Skillz - [Video] Use Bjorn's Super Guide to aim the ball so it will bounce off of a peg, and into the bucket. Make sure to compensate for travel time of the ball, otherwise you'll miss the bucket!
Duo-Slide - [Video] Use the Super Guide to bounce off of each of the pegs to do an extreme slide on the blocks. The Super Guide will turn fiery (flare) when properly positioned for an extreme slide.
Ridiculoso - [Video] Slide from the bottom of the top right circle and the ball will continue into the left and then bottom right circles.
Big Skillz - Same as the Skills, but without the Super Guide. The middle pegs are the easiest to bounce off of, but if you need practice, go back to The Skillz and watch how the ball bounces off of the pegs.
Wall-off - Use the Super Guide to bounce a ball off of the wall (left or right side of the screen) and into a peg. Repeat the process three times.
Five Oh - Clear all of the orange pegs. I will almost always recommend using Gnorman for this, but if you don't have him yet that's fine, because this one is relatively simple, especially if you get a double extreme slide on the two slope block lines.
Ultraviolet - Build a combo with orange pegs, leaving a couple alone while you clear out the remaining purple pegs to increase your points. The trial will end once you break 500k points.
Down Low - Don't link orange peg bounces over long distances, try and clear out small areas of orange pegs at a time, or long distance bounce to blue pegs. Avoid landing in the bucket, and waste your extra balls at the end of the round to avoid the spare ball bonus. Avoid the purple pegs, especially at higher multipliers.
Target Practice - The middle two will generally combo if you hit the second from the top, otherwise, try and launch the ball when the bucket is beneath it so you have as many shots as you need.