The equilateral triangle is the first construction in Euclid, and it takes two circles. Given a segment AB, strike a circle at A through B and a circle at B through A, and call one of their crossings C. Then AC and BC are both radii, so both equal AB, and ABC is equilateral. That is the whole of Proposition I.1.
The triangle is constructible in the technical sense that matters. The Gauss-Wantzel theorem states that a regular n-gon can be drawn with compass and straightedge exactly when n is a power of 2 multiplied by a product of distinct Fermat primes. Fermat primes are primes of the form 2 to the power 2 to the power k, plus one; the only ones known are 3, 5, 17, 257 and 65537. Three is the first of them, so the equilateral triangle qualifies, and so does every doubling of it: hexagon, 12-gon, 24-gon and onward, obtained by bisecting arcs.
Other triangles are equally accessible. Euclid's Book I constructs a triangle from three given sides at Proposition 22, provided any two of them together exceed the third. Bisecting an angle, dropping a perpendicular and copying an angle are all triangle work.
Three points not lying in a straight line determine exactly one triangle, exactly one circle through them, and exactly one plane containing them. This is why the triangle is the natural unit for describing a surface, and why computer graphics decomposes everything into triangles: three vertices are always coplanar, so a triangle is guaranteed flat in a way that a quadrilateral is not.
The angles of a plane triangle sum to two right angles. That is Euclid I.32, and it depends on the parallel postulate. On a sphere the sum exceeds 180 degrees; on a hyperbolic surface it falls short. The triangle is thus the figure that most directly registers what kind of space it has been drawn in.