There are a lot of different kinds of Minimalisms, so my first suggestion would be to explore a bunch of different composers with extremely open ears:
Philip Glass - Personally, my favorite work is his opera Einstein on the Beach, but his string quartets are also great, and the piano etudes can be a nice introduction. His work tends to still operate within the universe of tonal structures and even, to some extent, tonal progressions, just with a huge amount of repetition. After enough listening, the repetition starts to not sound like repetition anymore, at least to me.
Steve Reich - His early work tends to focus mostly of the "phasing" of multiple otherwise identical elements. That is to say, he likes to explore patterns or melodies or chord progressions in which a copy (or copies) moves out of phase with the original. The classic examples are his tape pieces It's Gonna Rain and Come Out, but also check out some of his acoustic pieces like Music for 18 Musicians. My favorite piece of his is Piano Phase, in which two pianos are playing identical material, but gradually move away from each other in tempo until one is a single sixteenth note ahead. Then they move again until one is two sixteenth notes ahead, etc.
Terry Riley - I'm only particularly familiar with his most-famous work, In C. The piece has a fascinating framework wherein a group of musicians proceed through a collection of a bunch of musical fragments, but each musician decides how many times they want to repeat each fragment. The result is initial and final alignment, with a delightfully diffuse middle.
Louis Andriessen - If you can, definitely get a copy of Bang on A Can's CD called Gigantic Dancing Human Machine. I think there tends to be an especially visceral quality to his minimalist work, and a lot of people would disagree with my putting him in this list, but check out Hoketus wherein two identical ensembles on either side of the stage (or either speaker) bounce notes back and forth.
Pauline Oliveros - I especially like her piece Sound Patterns. Her music is especially drone-like and is built around meditative and fascinating ideas. Read some of her writings or lectures about "Deep Listening" if you can. She can find amazing sounds in incredibly unlikely places, such as the sound of vascular processes inside a tree sped up and pitched down so that they're audible. She also does some sculpture work that can be quite fascinating, and that generally includes amazing sonic elements.
John Adams - (Not to be confused with John Luther Adams, who should probably also be on this list, but I'm less familiar with his work). His most popular piece is definitely Short Ride in a Fast Machine, and it's a lot of fun, but I especially like his massive piece Harmonielehre. Extremely exciting and often quite visceral, Adams' music is often seen as the Neo-Romantic wing of American Minimalism, and, as much as all these labels can be infuriating, I get where that description is coming from. He uses powerful melodic statements that achieve a kind of soaring quality that can be really inspiring.
Arvo Pärt - On the opposite extreme, Pärt is sometimes described as a "Holy Minimalist" because his work is often austere and based on religious subjects. Absolutely gorgeous harmonies. I would especially recommend his popular piece Fratres, particularly in the cello ensemble version. He developed a very specific contrapuntal technique called "Tinntinnabuli" or "Tinntinnabulation" that involves pairing a modal, largely conjunct, melodic line with one or more "Tinntinnabuli" voices that follow the same general contour but arpeggiate a particular harmony, usually a minor triad.
EDIT TO ADD:
I should point out that, beyond a very general vibe, it's hard to point to any particular common minimalist techniques. Each composer tends to have their own method of exploration, and they, to a large extent, don't think of themselves as belonging to any kind of single school. Some explore repetition (often with a goal of exposing tiny differences either in performance of listener perceptions that render a repetition new again) [Glass, Reich, Adams], some explore drones [LaMonte Young, Oliveros, Pärt], some explore compositional restrictions [Luther Adams, Andriessen]. Aside from Reich's "phasing", Oliveros' "Deep Listening" and Pärt's "Tinntinnabuli", I don't think there's as much to explore in that direction as there might be in, for example, Serialism. A lot of the minimalist composers were specifically reacting against what they saw as an over-reliance on procedures and methods.
The general sentiment of the above paragraph is true, but Robert Fink's answer makes a strong point about some general techniques that are relatively common. Great examples.