First off, let me just say I'm no golf pro, but I do have some experience with clubs.
In general, I'd agree with your coach/teacher. Club fitting is a great tool for helping people with their game. If you choose that path, a good fitter might charge around $150 for the fitting plus the cost of clubs, which is probably $400 (total) or so more than the normal, stock clubs. This may seem steep, but considering if a good job is done, you won't need any new clubs for a while, it's definitely worth it.
However, the problem that I would talk with your teacher about is your age. I'm younger like you and am still growing, despite being over 6'. Because of that, my swing, and probably yours as well, will change in the near future as a result of more strength, height, and a few other factors. This can completely change the fitting that you would have. Since I'm taller, what I do is I find a stock brand I like and get the 1" increase on them and customize for good grips. This usually costs only around $200 tops for a full iron set and shouldn't be much more for a full set.
As to the place to buy the clubs, I'd suggest a major retailer like Golfsmith. A bigger chain will make sure the clubs aren't fake, and though the prices may not be the best, they will be worth it. Plus, some, like Golfsmith, have launch monitors so you can actually see how far you hit it, your swing speed, and enable you to try multiple drivers for best results for free.
For the clubs themselves, I'd get the best wedges, putter, hybrid, and 3-wood or driver, depending on which you feel more comfortable getting in the fairway. Don't back off on the iron set though; I only say focus on these clubs because at your skill level, you'd probably hit 4-8 GIR per 18 holes. Of course, if you have the money and time, make sure all the clubs are the best in your mind. For the longer clubs, it's all preference, and I'd suggest the big name companies (Nike, Titleist, Taylormade, etc.). For irons, Mizuno is a great under the radar company for irons, but if you like hybrids, definitely look at Adams or other hybrid iron companies. I use Nike, but that's just preference. As to wedges, I've found that there's an elite group, an average group, and the rest. The Cleveland CG series (CG-15, etc.) and the Titleist Bob Vokey wedges are in the elite group. Most other big-name wedges (Callaway, Ping, Nike, etc.) are average. For putter, splurge as well. I like Odyssey, but Titleist Scotty Cameron and other big name companies, like Nike and Ping, are great.
For the prices, a good driver should be about $200-350, a wood for $150-200, a hybrid for $150-200, an iron set for $500-900, 2 wedges for $80-120 each, and a putter for $125-200. This totals for a conservative number of $1285 to a more pricey number of $2090.
I hope this helps and good luck with your search!