If compressing videos is all you're looking to do, then HandBrake is easily the best tool for that. It's free.
https://handbrake.fr/But here's a warning, it's quite complicated at first glance. If you decide to use it, post again and we'll try to help you if you're not sure what to do.
As for online storage, I'm not experienced with that. I save all my videos locally on hard drives because backing up gigabytes or even terabytes of TV recordings to the cloud would take a looooooong time to do, and depending on your ISP you'll quickly hit your monthly bandwidth limit as well.