How to Determine and Verify Series and Parallel I am having a lot of trouble visually identifying which elements in a circuit are in series and parallel is there a formal mathematical way I do this with Kirchoff's Law?
 A: Yes
... but you may not like it. It can be very math intensive, especially when a little practice or other methods tell you the same thing with less effort. Let me first suggest some other methods.
For most undergraduate and high school physics courses, you can usually identify this by drawing a circuit diagram, and use the smallest sections of circuit to identify if elements are in series or parallel, replacing those with a element of equal resistance or capacitance, and moving on to the next section of circuit. Alternatively, some people start at a terminal of their voltage source, and "follow the current" through the diagram or circuit and figure out which elements are in series and which are in parallel. (If the current must travel down two or more paths, items on those paths are in parallel until those paths reunite.)
However, you're looking for a mathematical way, with Kirchhoff's Laws. This is one:

If you complete the circuit in an attempt to verify Kirchhoff's Voltage Law, but you mistakenly identify some element in parallel or series, you will find that the law is violated. If you've correctly identified all the elements as in series or parallel, the law will be verified.

This may work well if you have only one or two segments of circuit that you're unsure of, but with multiple sections / complex circuits this method quickly becomes unreliable. As our friends at wolfram alpha show, this will only work when you do not have resistors with imaginary resistances (that is, a resistor whose resistance is a very specific complex number). If you ever find a such a thing, you will end up with a circuit that you can't tell if it's in parallel or series under certain conditions.
Edit:
You'll also need to know the current going through each item in the circuit, if you want to use Ohm's Law to determine the voltage. If you're using a real-live circuit, you could just take a voltmeter to each item in the circuit. You could also use that voltmeter on small portions of the circuit to determine if items are in series or parallel.
It was also pointed out (by Floris) that many circuits can have complex values for their impedance. Floris also pointed out that things "can get messy" when voltage/current are not in phase. Thanks Floris!
Gaining an intuitive understanding of how to identify elements which are in series or in parallel is a very, very valuable skill. In any case, the aforementioned method will work for most problems you'll encounter at most undergraduate and high school physics classes. Looking at more resources, you should try Khan Academy, which has videos explaining basic circuits. You should try looking at people solving Wheatstone Bridges, or other types of bridges, which may help you develop a sense of when items are in series or parallel.
A: I don't know if this answers your question, but whenever I talk  about in-series and in-parallel resistors I like the water analogy.
In this analogy the battery is a pump that lifts the water from low (potential) to high  (potential). The electric current is the water current, and the resistor is a wheel or a constriction in the pipe.
So I have In-Series resistors when the same (water-electric) current flow through all the in-series constrictions ($I=I_1 = I_2 $) like this image:

In the same sense I have In-Parallel when the water current splits and then reunites  ($I=I_1 + I_2 $) .  The constrictions (resistors) between the split - reunion of the current are in parallel, like this image:
 
With some practice problems it becomes easy to identify them!
Bonus another image from the web where the resistance is displayed as wheel:

A: Yes. there is a very simple method to determine that. 
If the two components are connected in such a way so that they have exactly one common terminal and no other component is connected to that common terminal then the two components are connected in series. 
The two components are said to be connected in parallel when they have two common terminals
Any other question in this regard ? 
A: Identification of series connection
    i. There will be one common terminal points between two elements which are in series.
   ii. The common terminal between the two elements is not connected to any other components.
I find these points in this link. I think it'll be very helpful for you:
https://elucidatetechno.blogspot.com/2019/05/what-does-series-circuit-mean.html
