I'd like to expand on Matthias's answer a little bit. There is some unfortunate terminology going around. Being a *cofibration* resp. *fibration* is a particular property a map can have; for spaces Hurewicz cofibrations and Serre cofibrations (relative CW-complexes) are ones that get most use, and they have associated notions of fibrations; the concept of a model category mentined by Dan Ramras is in the background of this but I don't think your question warrants, or requires, getting out the big guns. For spectra, in the usual model structure, a cofibration will be a relative CW-spectrum, and $X → *$ is a fibration iff $X$ is an $\Omega$-spectrum.

Every map can be turned into a cofibration or into a fibration if you allow yourself to change your target resp. source by a weak equivalence. For spaces, the mapping cylinder construction will turn $f\colon A → X$ into a Hurewicz cofibration $A → (A \times I) \sqcup X/\sim$, and the path space construction will turn $f\colon E → B$ into a Serre fibration. Similar constructions exist for Serre cofibrations and Hurewicz fibrations, and in fact it's an axiom of model structures that such replacements must always exist. Also in spectra.

In other words, in the *homotopy* category, it doesn't make sense to say that a map is a cofibration or fibration.

A *cofibration sequence*, or *fibration sequence*, is a completely different animal, although there's of course a relation. The usual definition of a cofibration sequence is as follows: if $A → X$ is a cofibration, then $A → X → X/A$ is a cofibration sequence, and any $K → L → M$ weakly equivalent to it is a cofibration sequence, too. So the first map need not be a cofibration. In fact, any $A → X$ can be extended to a cofibration sequence (by the mapping cone, to be explicit in spaces). Analogously for fibration sequences.

It is true in spectra that if $A → B → C$ is a cofibration sequence then it is also a fibration sequence, and vice versa. As a special case (choose $B$ to be a point), the loop functor is inverse to the suspension functor. Morally, in spectra, you've inverted the suspension functor $\Sigma$, so both $\Sigma^{-1}$ and $\Omega$ want to be right adjoint to $\Sigma$, so they have to agree. That's not a proof, though, but you'll find explicit proofs in any textbook on stable homotopy theory, e.g. Adams's "Stable homotopy and generalised homology", part III. The most conceptual reason of this is the *Blakers-Massey theorem*, which is a theorem about pushouts/pullbacks of spaces, which roughly says that a pushout diagram with highly connected maps is also a pullback diagram in a range of dimensions. The Freudenthal suspension theorem mentioned by Matthias is a corollary of this.

To address your more specific questions: $\Sigma^\infty$ preserves cofibration sequences, $\Omega^\infty$ preserves fibration sequences, but not vice versa. In fact, $\Omega^\infty$ turn cofibration sequences of spectra into fibration sequences of spaces, which is not surprising since cofibration sequences of spectra are fibration sequences after all.

And for your concrete question: you have a natural map from the homotopy cofiber of $X → Y$ to $Z$ whose connectivity depends on the connectivity of $X → Y$; the Blakers-Massey theorem will tell you the details.