Convert video file (.ts file to mkv)

I would like some to take a video .ts file that I provide and convert it into a .mkv file (retaining quality or as much as possible but reducing file size) to my satisfaction and then document exactly how this did this and using what software so that I can follow it for future. The file will be about 1GB in size. I have a 10MB upload speed. Please let me know how long you think this will take you to download and convert. Any questions please feel free to ask.