I've got a set of Keplerian orbital elements $e_0$, $a_0$, $i_0$, $\omega_0$, $\Omega_0$, and $\theta_0$, and I'd like to get to a different orbit with orbital elements $e$, $a$, $i$, $\omega$, $\Omega$, and $\theta$. How do I calculate (a) the amount of delta-v I'll need for this maneuver or set of maneuvers, and (b) which maneuver or set of maneuvers I should make to do this optimally?
Before I get to answering your two questions, there are four additional parameters you haven't thought of. There the two epoch times at which those orbital elements apply, which I'll call $T_0$ and $T$ to match up with your nomenclature, the time $t_0$ at which the vehicle departs from the initial orbit, and the time $t$ at which the vehicle arrives at the final orbit. It takes a certain amount of delta-v to accomplish this transfer.
How do I calculate (a) the amount of delta-v I'll need for this maneuver or set of maneuvers?
Fortunately, the calculation of this delta-v does not require integration, at least so long as the problem is kept simple. So, keeping it simple, I'll assume that orbit are Keplerian. Suppose you pick a random $t_0$ and $t$ subject only to the constraint that $t>t_0$, and suppose the vehicle performs only two instantaneous burns, once to start the transfer at time $t_0$ and another to end it at time $t$. This is a boundary value problem. In general, finding a solution to boundary value problem is much harder than finding a solution to an initial value problem. Fortunately, there is an elegant, integration-free algorithm to solve for the
delta-v needed to at the start and end of the transfer. As Christopher James Huff mentioned in his answer, this is Lambert's problem.
How do I calculate (b) which maneuver or set of maneuvers I should make to do this optimally?
There is absolutely no guarantee that a random choice regarding $t_0$ and $t$ were optimal. To the contrary; the odds are very good that a random choice will be highly suboptimal. But now you are asking about optimality as well. There have been papers galore written in the past 350 years about optimization, and papers galore will continue to be published well into the future. The approach used by the Jet Propulsion Laboratory and other organizations to plan missions from Earth to Mars is to pick times $t_0$ and $t$ from values on a grid.
Each point will have a cost; you mentioned total delta-v as the cost function. There are other cost functions. Dollars, for example, is a the ultimate cost function. Translating delta-v to dollars is nontrivial. So once again, keeping things simple, I'll assume total delta-v as the cost function.
Every $t_0, t$ pair will result in multiple $\Delta v_0, \Delta v$ solutions to Lambert's problem. Pick the cheapest in terms of total delta-v. Then try another $t_0, t$ pair, and another, and another. Do this on a $t_0, t$ grid and you'll have a sampling of the cost surface. Projecting this onto the $t_0, t$ plane using contour plotting visualization techniques will result in a visual representation. The bulls eyes on that contour plot (the pork chop plot to which Christopher James Huff alluded in his answer) provides a nice representation regarding when it's best to depart and arrive.
