This is a really interesting question, because it seems hard to prove that numbers are irrational without also proving they are transcendental (roughly not the roots of a polynomial with integer coefficients). I do not want to focus on arguments about whether or not something is a proof by contradiction. Most proofs can be dressed up that way without seeming to unnatural, and conversely. I want to focus on how you go about proving something irrational. Eg Euler gamma constant, to take a notorious example.
Of course, if you look at the Cantor side of things we expect "almost every" number to be transcendental, and few indeed to be irrational but not transcendental. But it seems to be easier for someone who has been through the standard graduate school courses to prove a number transcendental than irrational. (I am not knocking those courses, just trying to indicate the typical knowledge of someone attacking the problems).
Nonetheless there are some standard approaches. A good introduction is in Proofs from the Book 3rd ed (be sure to get that edition, this entry has been substantially over the editions - ie ISBN10 3540404600, Aigner & Ziegler). The leading approach is integrals. The classic Book proof there is unfortunately not in the book, but is available on the web as a free pdf (you can get it by the standard searches): the rejigging of Apery's proof by Frits Beukers, Bull LMS 11 (1979) p268-72. There is also an interesting informal article he wrote 25 years later "Consequences of Apery's work on zeta(3)" (I think it is on Arxiv). Various people have tried to generalise it with less success than one might expect so far.
Then there are the tricks. Such as the classic one first published as an exercise (!) (well, that is an exaggeration, an exercise with hints), in those elementary math books by Yaglom^2, now available in English translation - chock full of beautiful stuff, often based on Moscow Olympiad problems [see ISBN10 0486655377m "Challenging Mathematical Problems with Elementary Solutions, Vol II", Dover, section X p22-24].
But are they really just tricks? Tricks often hide deeper things. It is worth trying to figure out why they work.
However, there is certainly no equivalent of Alan Baker's work when it comes to irrationality. But then there are few numbers which anyone really believes will turn out to be irrational but not transcendental.