As the others have said, "Uverse" is just another cable provider. you need to do some research comparing your current package with your other packages to see if you want to switch (be sure you understand any penalties if you cancel your current provider).
If you want to get a Roku device, you still need to have internet coming into your house. I had a Roku about 4 years ago (2009) so it was an older model than what they have now. I hooked it up to my older tv (tv was bought in 2004 and so is definitely not a flat screen). I used those red, white, and yellow cables to connect it up. When I first got the Roku device, I was living in a rental situation and had my computer and my tv in the same room and so I just hooked up the roku directly to the computer and to the tv. This allowed me to have internet and tv on one thing (I know you can get tv through your computer, but my tv screen was a lot bigger than my computer screen). However, when I moved into my house, I wanted to connect the Roku wirelessly. I did that and it worked for a little while (wirelessly to the internet, it was still directly connected to the tv) and then it quit. I did some research on the problem and a LOT of people had that problem with the roku device. So, I did not purchase another one. Instead, I purchased a Sony Digital Media device which I love. It has lots of "channels" - including Netflix (which I pay the $8.00 a month for, and hulu, and Amazon, and exercise channels, etc.). I don't use all of the channels, but I defnitely use Amazon. I am an Amazon Prime member, so I can stream some movies/shows for free.
When you are deciding what to buy, read the specs on the digital media device and see how it is connected. I bought this Sony one specifically because you did NOT have to have an HDMI cable. I have it connected to my old tv with the red, white, and yellow cables. And, it works wirelessly with my router.
Also, just FYI, I did buy a "Vizio Smart tv" for my exercise room. Basically, I wanted a flat screen I could mount on the wall, get tv and the internet features. So, it has a menu screen for netflicks and amazon - without having to buy the digital media device. I'm very happy with that set up as well.
Oh, and one more thing you should know, most BluRay players now have the internet features built into them. So, if you are looking into upgrading your tv or dvd player, if you get a blu-ray player, you DO NOT NEED a separate digital media device. I was slightly annoyed to learn this because I ended up buying a bluRay player for the exercise room so my kids would have an alternate location to watch videos and it turns out it had all the internet connection features- so I did not really need the "smart" part of the tv afterall. Ugh.
Oh, and when you research your digital device, the specs should say what it works with (Amazon, netflicks, etc). If you go to Amazon, I believe you can research there and see which digital devices are compatible with Amazon.
(also, just another FYI, my parents have a more current version of Roku and are 100% happy with it).
Another thing to consider is to just get a tablet. You can snuggle up in a corner and watch anything you want (netflicks, dancing with the stars) (amazon would be contingent upon which tablet you got). No football extra stuff on the tablet.
So, do your research!
Good luck,
L.