Arrow will sell you a great antenna: http://www.arrowantennas.com/arrowii/146-437.html , or you can google "tape measure yagi" and build an okay antenna.
Then you can get the transmitter and receiver together, in a package like the regrettable BaoFeng UV-5R: https://www.amazon.com/BaoFeng-UV-5R-Dual-Radio-Black/dp/B00... . This is a terrible radio. Their quality control is so bad that you sometimes commit a federal offense by pressing the transmit button—and for $100, you can't afford the test equipment to tell! So maybe use it as the receiver in a two-radio setup, and transmit with something like a Wouxun KG-UV9D.
A Kenwood TH-D72A is an excellent choice to do the transmitter and receiver jobs, but hundreds of dollars. A Yaesu FT-60 is a great choice too, but it won't have full duplex—so you can't be sure your signal is getting out without help.
To listen to a satellite you need even less: a crappy antenna and an RTL2832 USB software-designed radio: https://www.adafruit.com/product/1497
For all of these, you'll end up wanting miscellaneous coaxial cable and a bunch of ends. Hams tend to build up a library of crimpers and spare ends and such, so look to borrow from a friend or a club as you get started. The 75 Ω cable that's ubiquitous for television signals will totally work, but you'll have much less loss and much much less stress on components if you buy 50 Ω cable.
Why not try the tutorial at https://www.rtl-sdr.com/rtl-sdr-tutorial-receiving-noaa-weat... ? If that works, you can expand from there: the tutorial teaches you a huge amount about aiming at satellites and the physical work necessary to capture signals from space.
With handhelds being dirt cheap and a yagi cut from a measuring tape being an option, I don't think getting on satellite for $100 sounds undoable.