Seifert-fibered manifolds are the easy 3-manifolds! The JSJ decomposition + geometrization says that you can split every manifold along a small collection of tori into pieces which are either Seifert-fibered or hyperbolic. But the hyperbolic pieces are truly the place where 3-manifold theory gets crazy (and interesting): most 3-manifolds are hyperbolic!
One justification for this claim is as follows. First, knots/links in $S^3$ are split in a trichotomy: there are torus knots, there are satellite knots (which arise by "tracing out one knot around another", and so are in some sense reducible), and hyperbolic knots. Most knots (in an appropriate sense) are hyperbolic. A theorem of Thurston says that if you have a hyperbolic knot, Dehn surgery on $K$ can fail to be hyperbolic for only a finite number of slopes $p/q$; even better, it is conjectured that you can have at most 10 exceptional slopes. So a vast majority of the manifolds produced this way are hyperbolic!
So you could try doing computations on various hyperbolic manifolds, in particular, for various Dehn surgeries on hyperbolic knots/links. The Weeks manifold, the hyperbolic manifold with smallest volume, is a nice example (surgery on a relatively small link).
If you're doing this to get a better feel for WRT invariants, it seems like a good idea; if your goal is to do something novel enough to write a paper, I suggest you find a faculty mentor with similar interests to talk to about this; they will have much better suggestions than I do!