Your budget is modest, and that is both fine and understandable. Your primary goal should be buying the most versatile pieces of clothing possible. Don't go for flashy pieces like shirts with multiple plaids. You cannot wear those very often without it being very noticeable, thus it's not versatile. Instead, hit up a few stores in your price range (Macy's, TJ Maxx, Marshals) and buy some light colored dress shirts - white, varying pleasant shades of blue, perhaps light pink or light yellow if you insist on some braver colors. Make sure they fit, as in you buy a proper neck and sleeve measurement. If there is a noticeable gap, you need a smaller size. If the shirt looks like a deflated hot air balloon at your waist, opt for a slim fit shirt. If you can afford it, perhaps try a Brooks Brothers outlet.
Sticking to the above will get you the most mileage out of a limited budget. You might not be as flashy as you'd like, but overall, you will be happier and you will look the part.
As for the shoes, buy a new pair of brown/black cap toes when you get a chance and the funds. Keep in mind: rectangle = bad.
For what it's worth, I'm probably closer to your age than 95% of this forum.