This issue is an annoying one because you hit Install on an app, and then you see the downloading spinning, but doesn’t get you anywhere.

Fix Google Playstore stuck in Downloading

The fix that works 90% of the time for this kind of issue is to go to settings > Applications and then find the “Download Manager” applications or “Apps”.  Once in there, hit on Force Stop, and then hit on clear data and clear cache.  After you do this, you should go back to the PlayStore and try downloading the app again.  This should work.

If this doesn’t work for you, try going to Applications list in settings and clear data/cache for Google Play Store app.  You can also try clearing Cache/Dalvik from recovery.  Also, try deleting your Google account and then adding it again.

Hope this helps and good luck.