Here are some tools used regularly by folks around here:
Spreadsheets: LibreOffice Calc, Recline.js, Google, Ethercalc
Making charts: Datawrapper, Plot.ly, D3.js & NVD3, Chartist.js
Processing and transforming: OpenRefine, R-Fiddle, PyData
With free online apps like DataBasic, IFTTT and Import.io anyone - regardless of coding ability - can start making things with data right away!
Visit the Data Journalism Handbook to learn how data visualisation works, and dataviz.tools for more..tools. To learn to explore tabular data, check out the Analyzing data, Cleaning data and Spreadsheet Formulae tutorials.
Further lists of data wrangling tools can be found at the Opendata.ch wiki, the School of Data Handbook and Open Knowledge labs.
Feel free to share your own!