Is it good to learn measure theory before topology or the other way around? What would be the best way to go about learning topology and measure theory?
Does topology make analysis easier to understand?
Or does learning measure theory beforehand make topology easier to understand?
 A: I think learning topology makes analysis easier to understand: Say, in Euclidean space, a convergent sequence cannot have two distinct limits. In classical mathematical analysis one used to pick $\epsilon_{0}=\dfrac{|a-b|}{2}$ to deduce the contradiction, where $a,b$ are the limit values. But if we switch into a topological point of view, convergent sequence means eventually its elements lie into a neighbourhood of $a$, and also in a neighbourhood of $b$. Since in Euclidean we can make those two neighbourhoods to be disjoint, the largest radius to the disjoint neighbourhoods would be $\dfrac{|a-b|}{2}$, that is the reason why we can get a contradiction in assuming the eventually elements lying in those two disjoint neighbourhoods.
It is the Hausdorff property which explains why needs to pick $\dfrac{|a-b|}{2}$ (or smaller) to be a candidate, this is hard to see if we merely do the usual $\epsilon$-argument along with triangle inequalities, it is simply a mess. But with the aid of topology, everything is clear, the strategy makes sense.  
A: I would start with Topology because it is more fun. They could be taken together if the student is willing to put the effort in learning both.
A: You will need to define limits, for instance to state the dominated convergence theorem.
You could probably restrict yourself to metric spaces, but you will miss some important results of measure theory. Thus topology should in principle come before measure theory.
That being said, you could learn both theories together! Such an enlightening treatment is given in the first chapter of Rudin's book "Real and complex analysis", which I strongly recommend.
