Here are some recommended sources of data

GDP and Growth Decomposition

• International Monetary Fund's International Financial Statistics (IFS).
• Historical Data (from as far as year 0) from Maddison's Historical Statistics for the Global Economy.
• The Conference Board's Total Economy Database includes data on GDP, Population, Employment, Hours worked and Labor productivity since 1950 for 123 countries.
• OECD's Population and Labor Force Statistics includes labor data broken down by gender, age groups and sector.
• Some notes on how to do some growth accounting work (like computing labor's share) are available at Tim Kehoe's website. 

Trade

• United Nation's COMTRADE has data on country to country trade by different levels of aggregation.
• Christopher Magee's Bilateral Trade Dataset joint with information usually used for gravity equations.

Banking and Financial Sector

• IMF's Systemic Banking Crises dataset includes information on systemic banking, currency and sovereign debt crises for 1970-2010, including some information on policy responses.
• Reinhart-Rogoff's financial crises data (banking, currency, domestic and external default or restructuring, and inflation) for 70 countries from 1800.
• World Bank's Financial Development and Structure Dataset of indicators of financial development and structure across countries and over time (since 1960).

Micro Data: Census and Household Surveys (Internationally Comparable)

• University of Minnesota's IPUMS International collects and harmonizes census data from around the world.
• Luxembourg Income Study Database harmonizes datasets with income, wealth, employment and demographic information for a large number of countries.

Micro Data: Census and Household Surveys (USA)

• University of Minnesota's IPUMS CPS integrates CPS data making it easier to work with.
• University of Michigan's Health and Retirement Study (HRS) is a longitudinal panel study that surveys a representative sample of more than 26,000 Americans over the age of 50 every two years.

Micro Data: Household Surveys (Argentina)

• INDEC's "Encuesta Permanente de Hogares" (EPH) surveys households regularly. Nowadays it is done quarterly, covering all the country. Previously it focused only on Buenos Aires. Data online is only available from 2003, but data since 1974 can be obtained as well from previous researchers.

Education Data (USA)

• General: Digest of Education Statistics has a compilation of data on all levels of education. 
• College: Delta Cost Project analyses trends in money received and spent in higher education.

Note: Here I have only included databases I have come across and worked with. There are many more datasets worth exploring, but it is sometimes hard to know they even exist.