Ryan,
Yes you would need to buy a guitar as you would have to practice at home.
Yamaha would be a good choice, it is an established brand and the quality is top notch.
I don't know if they are still having sale, if they do, you can get a good one for around $350.
I am in Hougang.
The proficiency that you seek will depend on the kind of playing level that you want to achieve.
Singing and strumming simple song, you can do this in a month if you put in the practice time of an hour a day.
More intricate playing be it from reading musical score to playing by ear, would require understanding the fingerboard, chord construction theory, appregio technique, notes reading, scales etc. It may sound extensive, but you learn these as you progress with your lessons.
There is no short cut, practice and loads of enthusiasm are the ingredients to get good at this instrument, even then the learning journey never ends.
If you do decide to take up the lessons, I can go with you to recommend the guitar that you can buy.
You may not neccessarily need to spend $350, there are cheaper ones around.
Thanks,
Tomo