How can I use browser userscripts on my Android phone?

Google Chrome on Android so far doesn't support plugins, of which Tampermonkey is necessary to run userscripts, so here I recommend Firefox for Android. You can install Tampermonkey addon in Firefox and then start with your favorite scripts. Note that all userscripts are not compatible with Tampermonkey such as SE-AutoReviewComments for which you need
USI- Unified Script Injector addon.

Where can I find scripts to play with?

Stack Apps is a Q&A site for apps, scripts, and development with the Stack Exchange API. People discuss about developing around SE, and publish their projects there. You can find a lot of userscripts in the scripts section.

One of my scripts isn't working

beeshyams asked me this question before, but I wasn't a Firefox user then, so I installed Firefox fresh and installed that userscript, and found nothing wrong.

When you encounter some issues with userscripts, try removing and reinstalling them, as well as clearing/resetting your Firefox addon, and see if the issue resolves by itself. Usually this solves 99% of them.