Solving connectivity issues usually boils down to resetting your router or playing around with network settings. Other times, you'll be scratching your head for hours, with no solution in sight. What can help in these situations is eliminating the bits that are working, so you can focus on the ones mucking up.
There are many tools available for checking various aspects of your network connection, but a lightweight one from Google for Chrome, called "Connectivity Diagnostics", offers a good selection of tests to poke and prod common issues, including DNS resolution, port blocking and response times.
While not extensive by any means, Connectivity Diagnostics can serve as a starting point before launching into more complex tools and troubleshooting routines and being a Chrome extension, all it requires is the browser installed to get working.