If you'd prefer to get discounts ahead of time, check Broadway box or theater mania. Those are the cheapest options if you want to sit in the cheap seats. TKTS is the cheapest option for good seats because they discount off the top ticket price, so all the tickets cost the same. (So for example, when I went to American psycho my friend and I had similar orchestra seats, and I paid about $20 more for mine through Broadway box than she did through TKTS... And her seat was actually better than mine because it wasn't eligible for the discount code. But I could've gotten a rear mezzanine seat for a lot less than her TKTS ticket with my discount code.)
As for which show to see, I guess it depends on what you mean by modern. American Psycho is definitely the most modern show out there. It's very different and unusual, and the score is 80s synth pop. However not everyone enjoys it as it's pretty dark and twisted. I really liked it. It also has Sunday night shows most weeks so if you wanted to go on Sunday night it's one of the relatively few options.
My personal favorite right now (well second to Hamilton which obviously isn't an option for discount tickets) is Bright Star. It's a new musical with a bluegrass inspired score (but I don't think you need to be a huge bluegrass fan to like it). Steve Martin wrote the music. It's sweet and heartfelt. It's set in the 20s and 40s so it's not modern in that sense, but it is a new musical. You can listen to the music online for free. http://www.npr.org/2016/04/21/474678104/first-listen-original-broadway-cast-recording-bright-star (For non-spoiler songs that will give you a sense of the score without giving too much away, I recommend If you Knew My Story, Woah Mama, Asheville, and Pour Me Another Round.). I am addicted to the score and for me it's the kind of show I could watch over and over again. In fact I recommended it to a friend on Tuesday and even though I'm seeing it again in June, I couldn't resist joining her. (She loved it and said she plans to recommend it to everyone she knows who comes to New York.)
Another good option would be The Color Purple, which again is not a new story, but it's stripped down modern staging. The performances are amazing.
Finally Fun Home would be another good option. I loved it, although not as much as I know some other people on these boards did. Unlike the others, it is a modern story. I don't know that I'd call the score catchy but it is lovely. The story is emotional and touching and very sad. The staging is quite cool, as its in circle in the square which is in the round and the most intimate theater on Broadway. It's going to tour, but I think it will be different in the big touring houses.
Whatever you choose, enjoy!!
