Three things contribute the most to the quality of a recording, in order:
- The skill of the performer(s) being recorded
- The skill of the recording engineer(s)
- The quality of the equipment being used
Even though the equipment is the least important of the three factors, it can help a lot to have at least a minimum amount of decent equipment to get started.
The built in microphones on smart phones are usually not great, although sometimes one can get decent sound from them. Note that the speakers on smart phones are pretty much terrible, so if you're listening through the phone speakers to what you've recorded, you really have no idea yet how good or bad it is. At a minimum I would get a decent set of headphones to listen on, and ideally you would have a nice stereo system that you can plug the smart phone into.
An external microphone that will work for your smart phone might be a good next step. I wouldn't spend too much on one of these because it won't be useful except with your smart phone (usually) and a phone is pretty limited in terms of recording capabilities. You can either get a microphone that plugs into an analog headset jack or a USB microphone (if your phone can support USB). You might look at the accessories page on the web site of the company that made your phone to see what they recommend and sell.
Once you have some kind of decent mic, there is a huge amount to learn about recording. The most important ingredients to start with are levels and placement.
Pick a place to put your microphone near your amp, leave it there, and make some recordings at different volume levels. You should be able to hear the quality of the sound change with different volume levels.
Once you have a level you think sounds good, keep the level the same and move the microphone around and make different recordings. Again, you should hear some big changes in the sound with different mic placement. Try to find the combination of placement and level that sounds best.
That's just the first week or three of learning to record. There is a lot more, and there are fine gradations of just levels and placement that you can learn about over time as you develop your ear. If and when you have questions about the next steps, we'll be here.
Finally, the more you listen to your recordings, the more you might start to feel like your playing could be better or your guitar rig could be better. Daily practice and lessons are great for the first, and saving your pennies and investing in quality gear is good for the second, but don't rush out to spend a lot of money. It only makes sense to upgrade your gear when you have trained your ear to really hear the difference.