The software "Regina" by Ben Burton has the census of all triangulations containing 11 or less tetrahedra. There are more than 16,000 distinct manifolds in this list of triangulations. So if you're interested in getting a sense for what 3-manifold triangulations look like, this is a start.
It sounds like one of the main features you'd be interested in is the "Skeletons" feature and the "Composition" feature. This latter feature will help you recognise common sub-triangulations, such as certain standard triangulations of a solid torus (layered solid tori is the term).
http://regina.sourceforge.net/
You can also export the triangulations to Jeff Weeks SnapPea and compute and visualize a Dirichlet domain (if the manifold is hyperbolic) there.
The most actively maintained version of SnapPea is Culler and Dunfield's SnapPy, which is available here: