How to get the best internet possible for my new gaming PC? DETAILS BELOW?

So I recently invested a lot of money in a new gaming PC build. In fact, all of my parts have already arrived except for one, which is my motherboard. With this PC build, I want to play many online steam games such as CSGO, H1Z1, Rocket League, Minecraft, and many others at high quality and speed. I also plan to edit and upload high quality videos to YouTube and also stream at decent quality on Twitch.tv. Trust me when I say this, the computer hardware will not be the issue. I purchased very high quality parts and I'm very confident in my gaming PC build. The only problem that I'm afraid will have a negative impact on my experience is my internet connection. At my location, my ISP is Fairpoint Communications, and the speed is decent at the moment but I'm looking to upgrade very soon hopefully. At the moment, optimum speeds at my location are around a 3-4 download, 1 upload, and a 40 ping. Yeah, I know, that sucks. But like i said i plan to upgrade very soon if possible. I live among a family of 4, and we all share one router and modem connection. As a family, we use multiple gaming systems, mulitple computers, and multiple phones, often all at the same time. We have a DSL connection and our internet is provided through our phone line. There's even a chance that our router and modem hardware might be out of date. What are my options for getting the optimum internet connection? Should I try to upgrade my internet service? Should I get my own router / modem?