There is more than one way that derived algebraic geometry generalizes ordinary algebraic geometry. The new affines don't help you much with quotients, which are (homotopy) colimits, but they give you well-behaved intersections, which are (homotopy) limits. On the other hand, you can consider functors from affines (new or old) to a category like simplicial sets that has better quotient behavior than plain sets. This gives you a notion of derived stacks, and I believe they behave well under many colimits.
I'm not sure what you mean by "reckless abandon". I tend to make mistakes when I'm not careful with my mathematics, even if I'm looking at derived algebraic geometry.
I'm not quite sure what kind of answer you're expecting, but here is a geometric example that may help to grasp some intuition.
In differential geometry, when an intersection is badly behaved (e.g. it doesn't have the expected dimension) one can geometrically perturbe one of the two factors. For instance, if you are intersecting tow submanifolds $X,Y\subset Z$, and if $X$ is locally given as the zero of some functions $X\overset{\text{loc}}{=}\{f_1=\dotsb=f_k=0\}$, you may want to introduce a deformation $X_{t_1,\dotsc,t_k}$ of $X$ defined as
$$
X_{t_1,\dotsc,t_k}\overset{\text{loc}}{=}\{f_1=t_1,\dotsc,f_k=t_k\}.
$$
One of the main idea of derived geometry is to replace these deformation/perturbation parameters $t_i$'s by a homological perturbation:
$$
X_{t_1,\dotsc,t_k}\overset{\text{loc}}{=}\{f_1=d\tau_1,\dotsc,f_k=d\tau_k\},
$$
with $\operatorname{deg}(\tau_i)=-1$ (my degree convention is cohomological).
Homological perturbation has two advantages above geometric perturbations:
it can be made functorial.
it exists even in the (quite non-flexible) algebraic setting, where geometric perturbation may not exist.
Let's try to apply informally the above reasoning to the case discussed in Jon Pridham's comment: consider $X=Y=\{x=0\}$ inside $Z=\mathbb{A}^1=\operatorname{Spec}(k[x])$. You deform $\{x=0\}$ to $\{x=d\tau\}$ and then proceed with the intersection of $\{x=d\tau\}$ with $\{x=0\}$, and get $\{d\tau=0\}$, which is just a ($k$-)point (it is $0$ in $\mathbb{A}^1$) together with a self-homotopy (given by $\tau$). This is indeed the "space" of derived loops in the affine line that are based at $0$.
I apologize for self-promoting, but you can read an informal account of how to view derived self-intersections as some kind of based loop spaces in the introduction of Calaque, Căldăraru, and Tu - On the Lie algebroid of a derived self-intersection.
Best Answer
Here is an example from Bhargav Bhatt's talk "Using DAG" at MSRI last week. Needless to say, any mistakes are mine.
Theorem. Let $X$ be a coherent (quasi-compact and quasi-separated) scheme, let $A$ be a ring complete with respect to an ideal $I\subseteq A$. Then $$ X(A) \to \varprojlim_n X(A/I^{n+1}) $$ is bijective.
Before going into the proof, let us consider the case $X$ is affine. Then $$ X(A) = {\rm Hom}(\Gamma(X, \mathcal{O}_X), A) = \varprojlim_n {\rm Hom}(\Gamma(X, \mathcal{O}_X), A/I^{n+1}) = \varprojlim_n X(A/I^{n+1}) . $$
The idea for the general (coherent) case is to replace $\Gamma(X, \mathcal{O}_X)$ with ${\rm Perf}(X)$, the category of perfect complexes on $X$.
Slogan. Affine schemes have "enough functions". Coherent schemes have "enough vector bundles (perfect complexes)".
The second idea may be due to Thomason.
More precisely, we have:
Proposition. Let $X$ and $Y$ be schemes.
(a) If $X$ is affine, then $$ {\rm Hom}(Y, X) \to {\rm Hom}(\Gamma(X, \mathcal{O}_X), \Gamma(Y, \mathcal{O}_Y)) $$ is bijective.
(b) If $X$ is coherent, then $$ {\rm Hom}(Y, X) \to {\rm Hom}({\rm Perf}(X), {\rm Perf}(Y)) $$ is an equivalence.
We must specify what (b) means (here is where DAG enters the picture). We consider ${\rm Perf}(X)$ as the symmetric monoidal $\infty$-category of perfect complexes on $X$ (complexes locally quasi-isomorphic to a bounded complex of locally free sheaves of finite rank). The ${\rm Hom}$ on the right means the $\infty$-groupoid (space) exact $\otimes$-functors. So in particular (b) implies that this space is discrete. The map in (b) sends $f$ to $f^*$, the pull-back functor.
"Proof" of Theorem. We repeat the proof of the affine case, replacing rings with categories of perfect complexes: $$ X(A) = {\rm Hom}({\rm Perf}(X), {\rm Perf}(A)) = \varprojlim_n {\rm Hom}({\rm Perf}(X), {\rm Perf}(A/I^{n+1})) = \varprojlim_n X(A/I^{n+1}) . $$ Unlike in the affine case, the middle equality needs some justification, which I am not ready to give.
End remarks.
(1) I think Bhargav mentioned that an idea due to Gabber allows one to get rid of the assumption that $X$ is coherent in the Theorem.
(2) He also said that the above proof is the only one he is aware of.
(3) Reference for the above (thanks to the user crystalline):
Bhargav Bhatt Algebraization and Tannaka duality arxiv.org/abs/1404.7483.