$\def\PP{\mathbb{P}}$In a course where all varieties are quasi-projective (as in Shafarevich Volume I), I am trying to figure out whether I can justify talking about the total spaces of the tangent and cotangent bundles to a smooth variety.

The tangent bundle to a (quasi)-affine variety is fine. If $X$ is (quasi)-affine, embedded as a subvariety of some vector space $V$, then the tangent bundle of $V$ is isomorphic to $V \oplus V$, and $TX$ is a subvariety of $TV$.

If $X$ embeds in $\PP(W)$, then the tangent bundle to $X$ embeds in the tangent bundle to $T \PP(W)$, so the question is whether there is a sufficiently nice proof that $T \PP(W)$ is quasi-projective. The best I can do is the following: For $\bar{w}$ in $\PP(W)$, the tangent space $T_{\bar{w}} \PP(W)$ is canonically isomorphic to $\mathrm{Hom}(k \bar{w}, W/k \bar{w})$. For $(\bar{w}, \phi) \in T \PP(W)$, choose a lift $\vec{w}$ of $\bar{w}$ to $W$ and a lift $\tilde{\phi} \in \mathrm{Hom}(k \vec{w}, W)$ of $\phi$. Send $(\bar{w}, \phi)$ to $(\vec{w} \otimes \vec{w},\ \vec{w} \wedge \tilde{\phi}(\vec{w}))$ in $\mathbb{P}(\mathrm{Sym}^2 W \oplus \bigwedge^2 W)$. Then the projectivization and the wedge product cancel out our choices, and I believe I can show this is a regular immersion of $T \PP(W)$ into $\mathbb{P}(\mathrm{Sym}^2 W \oplus \bigwedge^2 W)$. This construction is too ugly to present; does anyone know a nicer one?

The cotangent bundle seems to be much worse. It doesn't even make sense unless $\Omega^1$ is reflexive, but let's say $X$ is smooth.

It is true that, if $X$ is affine, then the total space of any vector bundle $E \to X$ is affine. (Use the fact that affineness of a morphism can be checked on any affine cover, see for example Prop 7.3.4 in Vakil, and contrast with Jason Starr's answer here.)

Also, if $X$ is quasi-projective, then the total space of any vector bundle $\pi: E \to X$ is quasi-projective. (Proof sketch: Let $\mathcal{E}^{\vee}$ be the sheaf of sections of the dual bundle. For $L$ sufficiently ample, $\mathcal{E}^{\vee} \otimes L$ is globally generated. I think we can then embed $E$ in $\PP(H^0(X, L)^{\vee} \oplus H^0(X, \mathcal{E}^{\vee} \otimes L)^{\vee})$.)

But these arguments are clearly not for a first term. Is there some more elementary way to do it, or does this just wait until next term?