Re: Reset iPhone 5s without losing pictures or using computer
Download Google Photos and upload all of your pictures and videos to the cloud. It's a free service and it's unlimited if you choose "high quality" instead of "original quality".
High quality is up to 16MP pictures, which is higher than an iPhone can take anyway (not including panoramas), and 1080p videos, which is as good as or better than any iPhone can take before the 6s, so you're practically saving the original quality anyway.
Back them up on Wi-Fi so you don't use your cellular data and you'll be able to access them at any time with a data connection. If you ever get a new computer or absolutely need to on your current laptop, you can choose to download them all from Google Photos.