If you wait until 12:00AM UTC (so about 50 minutes), you can pick up the latest version of Ubuntu, as it is going to be released then. Tomorrow's release (16.04) is an LTS (Long-Term Support) release, meaning it will be supported by Canonical and AskUbuntu for a very long time.
To install it, you need either a DVD or a USB stick. You should use a USB stick that is at least 4GB. Download the Ubuntu ISO from the website (16.04 won't be shown until it is released), and burn it to the USB stick using Rufus. Don't follow the instructions on the Ubuntu website. The program suggested is terrible.
In the installer, you should eventually be prompted with some options on how you want to install Ubuntu. Choose the option resembling
Erase Hard Drive and Install Ubuntu to do exactly what that says. Let Ubuntu install, unplug the USB drive when it's finished and reboot into Ubuntu.
Remember, Ubuntu is more than a little different than Windows, but there are countless tutorials online and we on AskUbuntu are (mostly) here to help. When you ask a question, just make sure it is a clear and detailed as possible. Also do your research before asking a question, as it may already be answered. Questions you may find about relatively recent Ubuntu versions (I'd say 14.04 and up) should be recent enough to work for 16.04.
Good luck, and happy Ubuntu-ing!
As some people have pointed out, there are more lightweight alternatives to normal Ubuntu. If your computer ends up not running very well, it may be a good idea to try out Lubuntu or Xubuntu.