I remember doing lots of analysis like this at university! Ah, they where the days.
There are only two times a voltage is the same at two points is if 1)there is a straight line connecting the two points together and no components are in that line. 2) Where a resistor is in the line and no current is drawn. For example an unconnected battery which comprises a voltage source and series resistance (the internal resistance of the battery). The voltage on both sides of the resistor will be the same as the voltage of the voltage source.
The rest of the analysis, is simply understanding how to combine resistors in parallel and series. Combine the resistors together, simplify the circuit (but you have to do it right!), redraw the circuit with new effective resistances then you can analyse further until you get to the point where you can calculate currents.
1) Research potential (voltage) divders, there's only a couple of simple equations for these and previous posters have done a good job at pointing them out.
Two other further key rules: (Kirchoff's Laws)
1) Sum of potential differences around a loop must some to zero. (sum of potential differences across loads = sum of potential differences of voltage sources).
2) Sum of currents into a junction (or node) equals the sum of currents flowing out of that node. That is, you can't just lose current, if it goes into a junction on a circuit diagram, it all must come out.