Beyond board power, understanding multiple power supplies and loads in a circuit
You usually need a common ground but don't want to connect power rails, even if they are the same potential (don't tie two separate 5V rails together, for example).  There are some exceptions. You can communicate across isolated grounds using opto isolators and there are some regulators designed to be safely used in parallel, such as Linear Technology's LT3080.  Otherwise, there is not much math.  You just need to make sure you can provide enough current for each of you rails' load requirements.  The only other thing I would mention is that if you do use separate rails of the same potential, and communicate between them, make sure you can't violate any I/O specs of your devices.  If you are using two separate 5V rails, for example, each +/-10% , one rail could be 4.5V and the other 5.5V.  Outputting a 5.5V signal to an input pin with a 4.5V supply could potentially damage the IC.  This is why, when using the same potential across multiple devices that communicate with each other, it's usually preferred to use a single supply rail for all devices.  This isn't always easy to do when connecting kits, however, such as the Arduino to some external device.

Edit, if you use one supply, tie all loads to it in parallel.  If you were to tie two loads in series to a 5V supply, for example, each load wouldn't be getting 5V.  If the two loads were exactly equal, they'd be seeing 2.5V each.  You design the supply for the rated load, taking into account heat considerations.  1.5A is a large load which will generate a lot of heat with a linear regulator.  You'd need to make sure that A) your regulator can provide at least your maximum load (with some room to spare) and B) can radiate the heat generated by the voltage drop.  This means you may require a heat sink.  Almost all regulator's data sheets will tell you how much heat is generated for a given amount of power.  The National Semi LM317, for example, says it's TO-3 package increases 39 degrees C for each Watt of dissipation.  If you are regulating 5V off a 9V rail at 1 Amp, the package would be dissipating (9V-5V) x 1A which equals 4W, or 146 degrees C over ambient temp!  Well beyond the maximum spec'd temp.  This is where heat sinks come into play, or the use of a switching regulator which are much more efficient, but probably too advanced for your first project.  The other thing to be concerned about for regulators is dropout voltage.  The supply rail needs to be some amount of voltage greater than your regulated voltage.  This is all assuming you have a AC to DC power supply to give you your main power rail.  But if this is the case, why not just buy a regulated AC to DC supply at the voltage or voltages you require with the current capability you require.  You can buy cheap wall warts with regulated output voltages in most common voltages and with various current capacities.