Recording a good sounding acoustic has been among the hardest things I have tried to do.
If you are not getting a "good mic'd acoustic sound" don't be too quick to blame the guitar since it's cheap, or old, or whatever. There's way more that goes into getting a good acoustic sound than that old guitar. I found this out when I bought my Taylor a few years back. I had been using an old, beat-up, broken neck, Ibanez. When I got the Taylor, I thought... OK this will solve the sound quality issues.... uhhhh not not so fast there hoss..... It sounded better, and it played better, and didn't have that broken neck, but,,,,, there were other things at play that I had to learn first, to get a pristine, and acceptable sound quality from the acoustic guitars I owned.
There are so many aspects to the sound of the guitar that the mic pics up, besides the sound of the strings, and that varies depending on where the mic is placed and even with the mic you are using.
The sound of the body with the lower and mid frequencies is generally a sound I do not wish to have a lot of in the final sound. I tend to mic further away from the sound hole as a result. If I do get the mic in close, sometimes 6", it tends to be off axis from the sound hole down the neck a bit and I often need to cut the mids to get a decent sound.
It has been an ongoing lesson in patience and perseverance for me in many respects.
I find that the mandolin, while acoustic as well, mics up pretty nicely close in. I guess the body resonance doesn't have the larger space and therefore a bigger mid/low resonant sound.
With an acoustic guitar, it will always sound so much better with new strings. Even my old "beater" acoustic sounds good with new strings. I heard years ago, and they may actually still do it, in Nashville studios, they would change the strings on the acoustic guitars during the recording sessions after just 30 minutes to keep the very bright sound of the strings from fading. I think that's a bit over zealous but it makes sense. They also used very heavy gauges since picking leads wasn't done often.... mostly strumming. I think, that while the guitar plays a big part in the sound you can record, (obviously), a fairly decent off brand name guitar with new strings, good intonation, proper use of EQ in the bin, and proper mic setup can sound amazingly good.
As far as where to start of how to proceed. Remember the KISS principle. Keep it simple. I generally place the mic on the stand and have it generally about 18" out somewhere directly in front of and slightly above the 12th fret aimed off axis slightly to the body and not aimed at the sound hole. With a bit of EQ in the bin I can generally dial in a decent acoustic sound. That's my strumming set up.
For a picked acoustic lead, I like to move the mic in a bit closer.... 8" to 12" max and closer to the sound hole but again not aimed directly at it. This allows me to play a bit more gently and still get a good solid signal into the track.
The further out you move the mic, the less "body honk" you get but the more "room" you tend to get as well. One or the other.....
Of course, this is just a step on the pathway to learning more..... the main thing is to keep at it, keep working on mic placement, experiment with 2 mics, (which I have yet to do on any sort of a serious level) and experiment with EQ settings to discover that perfect acoustic sound.
I'm currently working on a new song I just wrote and this one has some strummed and picked acoustic so I'm working on lesson # 1248 on acoustic mic'ing.... or it might be #1348 ... I don't recall at this point. Point being... it's an ongoing experiment for better sound.
Have fun.