For finding new apps, I have a few different ways of finding new apps for my phone and tablet. I tried out Droid of the Day and found it to be very similar to one of the other apps that I already use, AppGratis. Both of these discovery apps highlight a new, free app every day.
Droid of the Day seems to search through the Play Store and through user recommendations for apps. If you find an app that you use and would recommend, then like it, and it might be the next app of the day.
AppGratis is slightly different, they get developers to discount apps (usually the apps of the day are free or 90% off) to get you to install and start using them. The catch is that they are only free for that day. If you need to re-install it or install it on another device, you have to pay for the app.
Another service that I use to find new apps is AppBrain. AppBrain touts itself as an alternative Android market. It features many of the same apps as the official Google Play Store, but also has better recommendations. More “if you like this app, you might also like _____”. AppBrain also claims to scan all the apps that you have on your phone and uses that list to recommend you more.
As for websites that I trust to recommend new apps, one that I really like is Tested.com’s Android App Roundup. Every Monday they publish a list of 3 or 4 apps that have come out recently that they really like. Other places that generally publish good apps to use include Lifehacker.com and Alternativeto.net.