You can't produce every ($\infty$,1)-category from a model category. The slogan is that every presentable ($\infty$,1)-category comes from a model category, and every adjoint pair between such comes from a Quillen pair of functors between model categories. The paper by Dugger on Universal model categories works out this formalism from the point of view of model categories. (A companion paper shows that a large class of model categories (the combinatorial ones) are "presentable" in this sense.)
(I say it's a slogan, but I'm sure it's a theorem; I just don't have a reference for you.)
Presentable ($\infty$,1)-categories are special among all ($\infty$,1)-categories; in particular, they are complete and cocomplete.
For instance, you can define the notion of ($\infty$,1)-topos in terms of model categories, since ($\infty$,1)-topoi are presentable, and morphisms between such are certain kinds of adjoint functor pairs.
In practice, one tends to "compute" arbitrary homotopy colimits as bar constructions, especially when you have a simplicial model category.
If $X:J\to P$ is a simplicially enriched functor, where $J$ is small, then you get a "bar construction" $B=B(*,J,X)$. This is a simplicial object in $P$, with
$$B_0 = \coprod_{j_0\in \mathrm{ob}J} X(j_0),$$
$$B_1=\coprod_{j_0,j_1\in \mathrm{ob}J} X(j_0)\times J(j_0,j_1),$$
$$B_2=\coprod_{j_0,j_1,j_2\in \mathrm{ob}J} X(j_0)\times J(j_0,j_1)\times J(j_1,j_2),$$
etc.
(Here "$\times$" really means the simplicial "$\otimes$"; if $P$ is simplicial sets, then it really is $\times$.) If $X$ is suitably cofibrant, then the realization $|B|$ of $B$ will be the homotopy colimit of $X$.
This bar construction I described above is really a special case of "use the projective model structure"; you can use a bar construction to an explicit construction of a projective cofibrant resolution of $X$ (typically under some hypothesis on $X$, such as that each $X(j)$ is cofibrant in $P$). In fact,
$$|B(*,J,X)| = \mathrm{colim}_J |j\mapsto B(J(-,j),J,X)|,$$
and there is a weak equivalence $|B(J,J,X)|\to X$, which is a true projective cofibrant approximation given some mild hypothesis on $X$.
The standard references are oldies but goodies: Segal's paper "Classifying spaces and spectral sequences," IHES 1968, and the "yellow monster": Bousfield & Kan, "Homotopy Limits, Completions, and Localizations," LNM 304.
Added:
When $J=\Delta^{\mathrm{op}}$, you can say something easier: the homotopy colimit of $X: J\to P$ is computed by the realization $|X| \in P$ (again, up to the cofibrancy of the objects $X(j)$). I don't know an explicit reference offhand, though everybody uses this fact; it may be in the two that I cited.
Best Answer
I'm not really sure what you are asking for. Colimits and limits are easy to compute in simplicial sets, because it's a presheaf category (as you say). But if you want "geometrical" intuition about simplicial sets (including "pictures" of joins, etc.), you want to know about the geometric realization functor from simplicial sets to spaces. In particular, the fact that geometric realization preserves colimits (it's a left adjoint), and that it preserves finite limits (essentially a theorem of Milnor).
The limit result is proved in the book by Gabriel-Zisman. (Goerss-Jardine mention the result in chapter 1, but don't seem to prove it.)