Starting to apply theoretical "data science" learnings to real-world data sets I've been reading a number of data science books recently (Python, R, etc), plus attempting a number of the MOOC courses, and so on, and the content is reasonably consistent: regression, classification, neural nets, and so on. 
I've happily worked through the examples, which make sense, etc., but the struggle is when I get to the day job -- I'm an analyst, trying to get the next level of rigour. The challenge I have is "I have a problem that looks like a round hole" and "the text book looks like a square block"... plus the real world data has holes, data collection issues and so on.  
For those of you who have e.g. done graduate courses, are ML experts, etc., what advice do you have on how to building experience in how to apply the different techniques you've learnt? 
At the moment, I look at something like Kaggle challenges and just don't know where to start. I recognise that there is something to be said for working out the problem class - classification, prediction, etc -- and then giving it a whirl, but have people found this to be the most effective route forwards?
NB I recognise that this may be against the question asking rules of Cross Validated, but hoping it gets by on the "subjective" path... Googling for "applying data science" or similar terms leads to a bunch of sites trying to get me to enrol in school next year
 A: Been there, done that...
The best advice that I can give you involves two or three things:


*

*Keep learning... as you learn more you'll start to ask the right questions about the data that you are analyzing. The focus is really on three things: What am I performing the research for (i.e. what questions am I trying to answer), do I have the right data to answer the question (and was it collected in an appropriate way that will provide internal and external validity to the results), and finally how do I analyze and report the collected data?


If you haven't been exposed to these, I'd recommend having a look at the Handbook of Parametric and Nonparametric statistical procedures by Sheskin and Experimental and Quasi-Experimental Designs for Research by Campbell and Stanley. I've read Campbell and Stanley in its entirety and made it through the vast majority of the most recent version of Sheskin.
Going above and beyond the coverage of these books, it is important to understand data classification (ex. is a variable considered nominal, ordinal, interval, ratio, or absolute scale of measure) because it has a profoundly important effect on the statistical procedures you need to use and the applicability of the statistical learning/machine learning algorithms that you use. Ex. you might never apply a continuous valued neural network to predict a nominal scale dependent variable.


*Pick a pet project.


You'll discover that if you set out to determine something that interests you (say expected temperature for July in Colorado next year), then you will face all of the necessary trials and tribulations of developing a research plan, collecting data, and developing appropriate analyses.
You'll have a lot of late nights and a lot of pain through the process, but there are a lot of people in the world who can help answer questions when you get stuck. I've found that trying to apply new methods to my pet projects have led to some interesting, and perhaps non-intuitive, breakthroughs.


*Be creative.


As a data scientist, you have to be creative to develop the right techniques to use to solve a problem. Simply consider the brilliance of the "bootstrap." When the method was developed, it was unclear that it would work at all and many academic researchers dismissed it straightaway. Nowadays we have methods like Random Forests that are highly dependent on research first postulated with the bootstrap.
Statisticians don't always agree on the "right" thing to do to analyze/solve a problem. In fact, I would say that most statisticians probably disagree once a problem moves beyond an elementary level.
