I am aware that Incompleteness Theorem poses a problem for mathematics because no matter how large the initial axioms you start out with are, there will be some calculations that you are unable to make (assuming you have a consistent system.) For instance Goldbach's conjecture - that any even number + 1 is an odd number - is impossible to prove.

It's not really a 'problem' for mathematics. Mathematics is getting along just fine (most mathematicians don't interact with the theorem at all). There are lots of results in mathematics that say such-and-such is impossible, and this is just one of them.

Also, not every case of independence has to do with the incompleteness theorems. For example, here's a very simple axiom system.

1. 0 exists.
2. If n is a number, then there exists a distinct number n+1.

From these two axioms, it's impossible to prove lots of things. You can't prove 2+2=4. You can't prove there are infinitely many numbers, or that every number has a unique successor, etc. This doesn't have anything to do with the incompleteness theorems; it's just a mundane case of one statement being logically independent from a few other statements.

So if someone discovers that a statement is logically independent from a set of axioms (like the Continuum Hypothesis from ZFC), it shows that those axioms are incomplete, but not necessarily in a way that involves the incompleteness theorems. A better example would be something like showing that the existence of a measurable cardinal isn't provable in ZFC, because the provability of such a cardinal in ZFC would allow ZFC to prove its own consistency, which directly contradicts the 2nd incompleteness theorem.

I've also heard Turing made the situation even harder for mathematicians by showing that you can't know which calculations are unanswerable and which are but I'm a little unclear on this.

It sounds like you're referring to the halting problem, which in some ways is analogous to the incompleteness theorems, though it doesn't make the situation "harder" in any sense.

If mathematics is just a formal type of logic, what were the wider reprocussions for logic and philosophy of Godel's theorem?

Well, one philosophical implication is that mathematics arguably isn't "just a formal type of logic".

Was Russell's "Mathmatica Principia" challenged?

Prinicipa Mathematica was challenged for a number of reasons. A chief reason being that it specifically aimed to show that mathematics could be derived from logical principles, but not everybody agreed that the starting principles it appealed to (such as the 'Axiom of Reducibility') were 'logical'. So while it was an interesting feat of systematizing mathematics in a formal system, it didn't necessarily achieve its philosophical goal of establishing logicism (and this is quite independent of the incompleteness theorems).

Do we emphasise consistency more now that we cannot reason indefinitely to larger conclusions?

There isn't any sense in which the incompleteness theorems say that 'we cannot reason indefinitely to larger conclusions.' The work on large cardinals in set theory is an example of that; investigating types of sets that require stronger and stronger axioms in order to prove. The incompleteness theorems show us why proving the existence of these large sets requires stronger axioms; they don't show that we can't reason about these sets.

Anyway, to answer your title question more directly, I would say the following are some philosophical implications of the incompleteness theorems (and of the method Godel used to prove them).

1. It arguably showed that Hilbert's program (or at least one version of it) couldn't be carried out. Hilbert wanted to be able to prove the consistency (and 'conservativeness') of formal systems using only 'finitistic' methods (methods that people from just about any philosophy of math could agree to being uncontroversial). This, in effect, was a strategy to get critics of some of the high-powered math of the time (e.g., math appealing to infinite objects and non-constructive methods) to shut up, since the program would show (by methods approved by the critics) that the high-powered math was at worst a convenient notation system that simply made otherwise-provable claims easier to prove.

2. One of Godel's core ideas in his proof was showing that statements about formal systems (e.g., statements about a formal system's consistency) are basically just arithmetical statements in disguise. I think this makes it more difficult to adopt a purely 'formalist' view of mathematics. For example, some naive formalists might say that there aren't strictly speaking any truths about arithmetic; instead, there are just truths about what theorems do or don't follow from which sets of axioms. However, what Godel arguably showed is that this distinction doesn't really make sense, because facts about unprovability are just certain kinds of arithmetical facts.