I don't have itunes/computer that's not just for work?

I'm getting a new iphone & wondered how I could save my videos from the iphone that I have now without those things? Can anyone help me find a solution?

You need a cloud service like Google drive to store them and then only view from that app, not the vids app. But check on if it's possible to get them there from the vids or camera roll section without a pc. Your iTunes is already installed on the phone so that's not a problem. Best luck.