First of all, dark matter and dark energy, despite their naming, are two very different concepts. We don't really have any good reason to group them together, other than the fact that both represent things we don't understand. Thus they are not necessarily backed by the same sets of evidence.
Why we believe these things exist
As it happens though, some of the strongest evidence for both comes from the cosmic microwave background. Basically, whatever "stuff" there is in the universe will have an effect on the temperature and polarization fluctuations in this radiation, which was emitted some few hundred thousand years after the Big Bang.
The best all-sky map of this radiation is made by the WMAP satellite, and every couple years they release several papers with the analysis of the data. For instance, here is the 2011 paper focusing on the cosmological parameters. They basically just feed all the data into an enormous statistical program to find the most likely values for a large set of parameters, including the amount of "normal" baryonic matter $\Omega_\mathrm{b}$, the amount of cold dark matter $\Omega_\mathrm{c}$, the amount of dark energy $\Omega_\Lambda$, and the dark energy equation of state parameter $w$. There is a lot that can be said as to what the effects of these parameters are, but ultimately you just cannot explain the CMB without having dark energy and dark matter.
Alternate theories for dark matter
Now "cold dark matter" (the CDM of the $\Lambda\text{CDM}$ model) means massive particles interacting via gravity and the weak force but not via electromagnetism and that were non-relativistic even at the time of recombination when the CMB was released. Just plain old "dark matter" refers to any gravitating mass that doesn't have much of an electromagnetic signal. The galaxy rotation curves you refer to were some of the first dark matter evidence, and indeed they could be explained by assuming a large number of quiescent black holes or star-less planets or dust that we just missed for one reason or another. These alternate theories could also, with enough manipulation, explain the bullet cluster, where the gravitational mass found with lensing maps is clearly not collocated with the baryonic mass in hot, X-ray emitting intracluster gas. However, microlensing surveys tend to rule out the first two, and we think we have a good handle on dust dynamics. Something more exotic is called for. There is a diminishing community in support of MOND - modified Newtonian dynamics - which postulates long-range deviations from the inverse-square law in gravity. However, the bullet cluster, together with very precise Solar system data, makes this theory difficult to get working.
Add to this the very nice "WIMP miracle" (no good wiki page there - sorry), which is suggestive of dark matter being new types of particles. "WIMP" stands for "weakly interacting massive particle," and the "miracle" is this: If you assume there is a species of particle $\text{X}$ whose only appreciable interaction is annihilation with its antiparticle $\overline{\text{X}}$, with a cross section typical of weak interactions and a mass typical of, well, particles, you can easily calculate the abundance of these things in the present universe. They are in equilibrium with other species when the universe is energetic enough for them to be pair-created, and they "freeze out" when the universe cools enough. The end result is right around what we infer from other means.
Alternate theories for dark energy
Dark energy is a little trickier. There really is not a good explanation of "what" it even is - the name is more a catch-all for describing the observed accelerated expansion of the universe. You might believe it is a cosmological constant. In this case it is a nonzero scalar $\Lambda$ in Einstein's equation
$$ G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi T_{\mu\nu}, $$
where $g$ is the metric containing all the information about the curvature of spacetime, $G$ is a known function of $g$ and its first two derivatives, and $T$ is the stress-energy tensor containing all the information about matter, energy, and momentum in the universe. This is equivalent to saying there is some substance in the universe with equation of state $P = -\rho c^2$ ($P$ is pressure, $\rho$ is mass density). [For comparison, non-relativistic diffuse matter is well approximated by $P = 0$, and relativistic matter has $P = \rho c^2/3$.]
Others are open to the idea that $w = P/(\rho c^2)$ is not exactly $-1$ for this new "stuff," and so that can be a free parameter in your modeling. All the evidence points to $w$ being consistent with $-1$, but the uncertainties are still somewhat large.
You can get more exotic and note that the accelerated expansion phase the universe is currently undergoing is not entirely unlike the inflation many believe happened in the very early universe. Inflation has all sorts of theories proposed for it, many of which are variations on "there is an abstract quantum field $\phi$ with these certain properties..." Others take place in the realm of strings and branes. Many of these theories can be boiled down to a $w$ that changes over time.
Future research
These fields of study are very much alive and well. Dark matter calls for both astrophysicists to study its role in large-scale dynamics and particle physicists to try to nail down its properties. The astrophysics side involves observers placing better constraints on the evidence so far, which has a lot to do with galaxy clusters and large-scale structure, as well as theorists to predict how dark matter influences e.g. galaxy formation, usually by using massive simulations. (And those simulations, such as the Millennium Simulation, often lead to cool movies.) The physics side involves experiments to try to detect the stuff directly (there are literally dozens of these - far too many to list - and though there are some claimed detections, none are really accepted by the community at large). It also involves finding a place for such particles in an extended version of the Standard Model.
Dark energy is begging for physicists to come up with testable models that mesh with the rest of physics without seeming fine-tuned. It is still very much new, given that it was only last year's Nobel Prize that recognized the teams that provided undeniable proof of its existence around the turn of the millennium.
So by all means feel free to be inspired by these problems to work in some field. There is plenty to be done. I would even say the prospects are better than e.g. string theory alone, since we have solid, incontrovertible evidence saying there are gaping holes in our knowledge when it comes to dark matter and energy, and these holes are right where we can easily probe them via astronomy.
