first off, welcome to KC Zane, and thanks for your service to our country. Your question about which courses to play is going to get a bunch of different opions on here. we all have our favs and like to tell you which ones are best and why, so take it for what it is.
If you dont want to do alot of waiting i would say, stay away from Rosedale on most days, unless you want to play the down under course there, which is one of my fav courses and its in the shade the hole time. plus its super short and super fun.
Drinking is allowed at almost all parks, but please dont bring glass bottles. And as for me personally, i buy all my disc golf stuff from disc golf world, and i would say they have the best prices, plus all the people that work there can answer any or most questions you have with all the equipment and can point you in the right direction with disc selection. We also have leagues just about every evening just check the schedual to find one to go to, and the bonus, when you play as an AM, you can win "funny money" and use it as disc golf world.
This all of course is just my .02 worth and im sure you can get better answers from someone alot smarter than me. My fav courses though, are Water Works, which is one of the finest courses you will find anywhere in the country, Cliff Drive is beautiful and right downtown and has a great mix of holes to work on your entire game, Swope would be a good beginner course and rarely has any kind of wait. We have so many great courses in the area you wont often be waiting to tee off, unless its rosedale. for some reason thats where alot of rec players play. And if you feel like having an insane time on a course, check out Blue Valley. Its SUPER long and its almost like a ball golf course, but has some amazing holes to see, and it also has a 9 hols short course. Best of luck to you and again, welcome to KC. Please check out a league sometime and meet some of the locals and get to know kc disc golf.