First off, welcome to Android Community, we are so pleased to have you here with us. Second, great question, I am sure you will get some great suggestions from our members.
One of my favorite free apps is Go SMS Pro. It replaces the native text messaging app on your phone, with a lot of great features. The app itself is free. If you want some of the more extravagant themes to go with it, then you will pay for them. But I find I am perfectly happy with the free ones.
I also love Yelp. It is great for helping me find local places and recommendations of where to go and what to try.
There are several good free weather apps as well. I personally use a paid one, WeatherBug Elite. But their free version, as well as the Weather Channel's free version are quite good as well.
Of course, if you are a social networking geek, like I am, you will want Google + and Facebook. I know I have to keep up with everything my friends are doing, no matter where they are.
I have many more, but need to get to work right now, unfortunately, duty calls. I will add more as time permits later on today.

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