I think you are correct to worry about an overload of algebraic structures, especially if they are not well-motivated. I would strongly encourage you to keep the naming of various algebraic structures to a minimum in a first course in abstract algebra. In my view, the goal of the course is for them to see the power of abstraction in a fairly concrete setting, illustrating how it illuminates certain structures (e.g. the study of symmetry via group theory, the similarity in structure between integers and polynomials in one variable with coefficients in a field via ring theory). I think it is much more useful to get to something non-trivial and engaging for the students such as the wallpaper groups, Platonic solids, or Rubik's Cube rather than trying to convince of the austere beauty of abstract algebraic structure for its own sake.
If you insist on introducing a bunch of different structures, my guideline for when it is appropriate to move on from one structure to another is when students, at a minimum, can do the following:
(1) distinguish between objects which have the structure and which do not. (e.g. is such-and-such set with such-and-such operation a quasigroup? is such-and-such subset of a group a subgroup?)
(2) be able to work with simple ideas related to the given structure such as substructures, quotient structures (though this is challenging in a first course) and homomorphisms. (e.g. is the following set-theoretic map a homomorphism of groups? is the following quasigroup a Moufang loop? what is the kernel of the following group homomorphism?)
If students do not become proficient at both (1) and (2) for a given algebraic structure, then it is not clear to me what is the point of introducing said algebraic structure in the course. As I suggested in my first paragraph, I am very skeptical that students can actually acquire significant mastery of multiple algebraic structures during the course of a semester. I think it is much more valuable to students to understand group theory at a deep level than to have a broad, but shallow, acquaintance with a wide variety of algebraic structures.