How to get convinced that there are a lot of 3-manifolds? My question is rather philosophical : without using advanced tools as Perlman-Thurston's geometrisation, how can we get convinced that the class of closed oriented $3$-manifolds is large and that simple invariants as Betti number are not even close to classify ?
For example i would start with :


*

*If $S_g$ is the closed oriented surface of genus $g$, the family $S_g \times S^1$ gives an infinite number of non pairwise homeomorphic $3$-manifolds.

*Mapping tori of fiber $S_g$ gives as much as non-diffeomorphic $3$-manifolds as conjugacy classes in the mapping class group of $S_g$ which can be shown to be large using the symplectic representation for instance.
I think that I would like also say that Heegaard splittings give rise to a lot of different $3$-manifolds which are essentially different, but I don't know any way to do this. 
So if you know a nice construction which would help understanding the combinatorial complexity of three manifolds, please share it :)  
 A: Read Chapter 4 of Thurston's notes http://library.msri.org/books/gt3m/. He produces infinitely many closed hyperbolic 3-manifold of different volumes, and hence non-homeomorphic, just by doing Dehn surgery on the figure 8 knot in $S^3$. This does not used advanced geometrisation theorems, just some basic (but clever) hyperbolic geometry, combined with the Mostow rigidity theorem.
A: I'm not sure if this is what you have in mind by "without using advanced tools" or "a nice construction which would help understanding the combinatorial complexity of three manifolds," but how about identifying pairs of faces of different polyhedra?  I guess this is what convinced Poincaré...
A: Here are two examples  suggesting the complexity of the  world of $3$-manifolds.
The first is the classical result  that any  $3$-manifold   can be obtained by integral surgery on a link in $S^3$.  If you believe that knots and links  form a complex Universe, than this result should suggest that $3$-manifolds    cannot be much simpler.
The next example comes from the striking work of  Dunfield and Thurston  on random $3$-manifolds.   You can get such things by picking  random elements  in the mapping class  group, where randomness is generated by a random walk on this group.  This has lead to the discovery  of  strange $3$-manifolds. For more recent  work on this topic see also this paper of Lubotzky, Maher and Wu.
A: Maybe by looking first at homology $3$-spheres and in particular to Brieskorn manifolds $M(p,q,r)$:
the link of the singular point $(0,0,0)$ of the hypersurface
$$z^p_1+z_2^q+z^r_3=0$$
with integers $p,q,r\geq 2$. 
Milnor studied these manifolds in his beautiful paper "On the 3-dimensional Brieskorn manifolds M(p,q,r)". in: Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox), pp. 175–225. Ann. of Math. Studies, No. 84, Princeton Univ. Press, Princeton, N. J., 1975.
A: This does depend (as others have said) on what you mean by "advanced tools", but what about looking at Seifert-Fibred manifolds? These are ones that are locally products $S^1 \times D^2 / C_n$ for some cyclic group $C_n$. Alternatively, you can view them as appropriate notions of a bundle over an orbifold surface $S$.
It's worth noting that, as I recall, a large variety of three-manifolds can be Seifert-fibred. But even ignoring that, you get the fact that you can construct them easily from an orbifold surface (quite a concrete object) together with some extra discrete data.
A: I would explain it like this:
Start with a framed link in $S^3$ and do Dehn-surgery along it to get a new $3$-manifold. In fact one can get every compact oriented $3$-manifold in this way as mentioned earlier.
To distinguish this $3$-manifolds one can use the fundamental group. Actually the fundamental group can distinguish almost all $3$-manifolds. It is easy to give a presentation of the fundamental group out of a surgery diagram:
First use the Wirtinger presentation of the link exterior and then add a new realtion (of the form $p\mu+q\lambda=1$) for every surgery.
Of course its difficult to distinguish two such presentations in general. But to convince someone that there are many different should not be hard. For example one can do the following:


*

*make this groups abelian and use the classification of abelian groups.

*compare the orders of the groups (this is how Poincaré proved the Poincaré sphere to be not $S^3$).

*show that some of this groups are non-abelian (for example by finding surjective group homomorphisms in non-abelian groups) others are abelian.

*compare orders of the elements.

