Rye has a distinctive spice to it, Jake. I was into ryes a lot early in my bourbon/whiskey journey, but drink less of it now. I'd suggest starting with something basic and inexpensive like Sazerac to see if you like the rye notes. I think ome stores even carry a 200ml bottle. The ryes I enjoy now are Whistlepig single or two barrel blends with aged Alberta rye. Not cheap though.
Some under $60 bottles I've enjoyed recently:
Old Forester 1920
Knob Creek 9 year
Early Times Bottled in Bond
Jack Daniel's Bottled in Bond