If my planet has two moons, assuming they have separate orbits, is there a "typical" equilibrium that they are likely to find? Does a moon of a particular size tend towards a certain distance or orbital period, or are these more or less arbitrary depending on other parameters of the system?
There is no standard behavior for the orbits of multiple moons.
You've uncovered a problem that has plagued physicists since the 1600s when Newton was beginning to describe gravitational mechanics. This problem called the 3-body problem or more generally the N-body problem. To quote Wikipedia the N-body problem is described as:
Given the quasi-steady orbital properties (instantaneous position, velocity and time) of a group of celestial bodies, predict their interactive forces; and consequently, predict their true orbital motions for all future times.
This is easily solved in cases of two objects (a planet and 1 moon). For all other cases there is no easy way to predict the future movements of every object.
The size of the moon, as a celestial body, doesn't matter much in the orbital equations, stable orbital distances are determined, almost entirely, by the speed of the orbit alone. The Jovian moons do suggest that there are certain Harmonics that tend to emerge in multiple moon systems but that's one example only we don't have hard and fast rules for such things. Looking at the mechanics of Lagrange Orbits might give you some insight into the relationships between moons that may naturally develop. Do bear in mind Roche Limits when putting together your numbers.
If the moons are very small and don't have a significant gravity field of their own, they are basically independent, e.g. Mars' moons. If they are large enough to have a significant gravity field of their own, two moons or more will perturb each others orbits. This leads to one of the moons being ejected from the system. There are exceptions, such as some orbital resonances as mentioned by Ash, but those depend on the size of the host planet compared to the moons. If the planet is much larger than the moons, orbits can be stable, e.g. if they are in resonance (e.g. the Jovian moons). I think multiple large moons would require a stable resonance for the system to be stable, but I'm not sure about that.
If the planet is not much larger than the moons, for example as with the earth-moon system, a stable system becomes very difficult, and likely all but one of the moons will be ejected.
If moons are in resonance that means their orbital periods are simple ratios, e.g. 1/2 or 3/5. So that is an in some cases standardised behavior, but the absolute orbits can be anything, there is no standard behavior for that. Also, if a moon and a planet are very close together, strong tidal forces will result, which will over time lead to tidal locking: the bodies will rotate around their axes in the same time that it takes to orbit each other. Pluto and Charon are tidally locked like that. The moon is also tidally locked to earth, which is why we only see one side of it. The earth is much larger so this hasn't happened for earth.