Well, I hate to break it to you, but you are NOT likely to get better audio with software. You are going to need some hardware to do this. Laptop analog audio outputs are typically pretty low quality.
However--You do mention MP3 files. That is likely a large part of your problem too. Do you know the quality of your MP3s?
Step 1: Does your laptop have a CD drive? If it does, then you need to play a CD off the CD drive and see if the audio is much improved. If the audio suddenly sounds great, then you likely have poor quality MP3s as the root of your problem.
If the CD still sounds bad, then we need to keep going to find the issue. This will likely mean that you need some hardware (i.e. a USB audio interface or a DAC). The idea here would be to pass the audio out of the laptop in the digital domain before doing the D/A conversion.
What headphones are you using?
Yes I have mentioned mp3s, but regardless of the file types, the files themselves sound worse when listening to them through the the "laptop analog audio outputs," as you mentioned-- rather than what there is on an mp3 player-- I guess their "mp3 player analog outputs" which are maybe designed more with music in mind? Seperately, I know there are different internal processes that relate to the final result of what you hear but I don't know enough about these processes to know what errors can happen or what the likley culprits might be for a reduction in sound quality.
Thanks for your patience in responding to the question, and thank you for mentioning the "laptop analog audio otputs" are typically pretty low quality, which I did not know, that is an area now open to me for further troubleshooting, "workarounds," that would be worth looking into when I can afford additional equipment.
You have actually mentioned now I see two things I could try to override or "bypass" the built in outputs that you say are usually low quality, and that I may as a result be hearing. I hope to become informed about technology and audio quality.
Thanks also for the troubleshooting recommendation. I do have a CD drive, but I lost every one of my CDs so I can't put a CD in my computer. It might be a good idea to invest in a familiar CD, simply to begin troubleshooting the system.
I don't know how one would pass the audio out of a laptop still in digital form as you said, but I am going to assume that is taken care of by the DAC interface and won't require removing computer parts as I have heard people do with their iphones. I had never heard of a USB audio interface before, so thanks for mentioning these things.