Should I start with correlations first before doing any testing? I'm using open source data to analyze innovation in a business context for my thesis. I've got the questions from the survey in an Excel file and although there's only 13 questions there's over 600+ variables and it is confusing for a beginning SPSS user like me. I want to see if there's a relationship between "innovation optimism" and the "environment" of a firm's home country (Japan). 
Is it best to start with looking for correlations? I'm not sure how many variables I can select at one time to test this. (i.e., if bivariate correlations only let me test 2 variables at a time how do I test multiple ones to see if there's a correlation?) Also, how do I select only Japan and not all the 22 countries surveyed?
 A: OK, first some (unashamedly subjective) advice about how to proceed.  I'll address your specific questions during and after this bit.
Correlations

"Is it best to start with looking for correlations?"

No, it's best to start with some theories that connect 'innovation optimism' and 'environment'.  Any theory about those concepts require some way to measure them, so first need an operationalisation in terms of the survey questions.  Then you can connect the theory to your data and get to the testing part.
Operationalisation

I've got the questions from the survey in an Excel file and although there's only 13 questions there's over 600+ variables

I'm going to guess that your variables are a mix of question answers, demographic information about the respondent, survey metadata, and derived quantities that people tend to leave in such files.  (In a worst case your questions are Likert items and generated in SPSS as 5 yes-no variables, one for each level.  Better turn those into something ordinal first.)
First you need to known how to characterise 'environment'.  That might be as easy as a country indicator, in which case it's already there for you, but it might be more complex, e.g. something involving business sector, company size, etc.  That depends on what you (or someone else) theorises are the causally relevant features of environment (e.g. one might say it is the national business culture, another might say that business size is more relevant).  You may need to construct one or more new variables to capture these characterisations.  Or not.  (In any case, here's another reason not to correlate: environment as you describe it is probably a nominal variable and the default correlation measures assume it is not).
Then you need to characterise innovation optimism.  Hopefully you have enough attitude questions to create an optimism index by combining suitable responses.  If you do it right that'll work better than looking at individual questions and give you one single continuous variable to work with.  There's lots of questions around this site and on the web about making indexes from survey responses.
Then you need to decide what else would, theoretically, cause changes in innovation optimism. These are controls, because you want to know the effect of environment itself, not of the bundle of other factors that come along with it.  (btw this is why you don't leap directly to correlations, because these don't even try to control for these other causes and so can be very misleading.)
By the time you're here you actually don't have 600 variables, but more like a handful that are relevant to the problem.  
Modeling
You are now in a position to fit a multivariate model to predict optimism on the basis of environment while controlling for other factors.  A good regression text will help you do this (SPSS's own manual entries are not too bad actually, although it's notoriously easier to run a regression than to understand the results...)
Questions

if bivariate correlations only let me test 2 variables at a time how do I test multiple 
  ones to see if there's a correlation?

With a regression model.  (Pearson) correlations look for the amount of linear relationship between two continuous variables.  Regression models such as the one I'm suggesting look for the relationship between different environments and average levels of optimism while correcting for other factors.  
If all the variables were continuous the methods are very closely connected, but you should test things using a model of your actual research problem - one that reflects one or more theories about it - not using a general method of measuring association between two things, that doesn't quite fit.

Also, how do I select only Japan and not all the 22 countries surveyed?

If I understand the research question then I think you don't want to.  I think you want to characterise 'Japan' (a proper name) in terms of its particular combination of environment and other variables (a description), just like you do every other country, then look for the general effect of environment across the countries surveyed.
Alternatively, if there is variation in the environmental variable within Japan, then you can indeed just toss out the other countries and work with a smaller data set, knowing you'll have some variation to explain optimism. 
There are a large number of more sophisticated approaches to such regression models, some of which are mentioned by @Michael-Chernick below, but before getting to them I think it is important to get the shape of the research process down first.
Hope that helps.
A: I would start by generating a sample correlation matrix for a large subset of those variables.  Eliminate the variables with low correlation and test the one's that appear to have high correlation.  This selection process technically violates the assumption of independent testing.  But I think it is still a reasonable approach as long as you take multiplicity into account. If you only want to focus on Japan then look at only the Japanese data.
However, sometimes in estimation problems however the estimation of parameters for one country can "borrow" strength from the results for other countries. In such cases empirical Bayes methods can be used.  In their baseball player batting average example, that is how Efron and Morris use other players' batting average to improve the estimate of a given player's average.  So there could be an advantage to using all the data.
