@DrDiettrich: Thank you for taking a look at those links. Since I am looking at them from several different companies, I can understand the possibility of not having sufficient data to link these parts. If I were to buy all components from the forum store/adafruit, would I have a better chance of finding parts that match? Could you explain what I should be looking for on the data sheets to check for compatibility?
Most important is the operating voltage of a module, as far as logic signals (data lines) are affected. When it differs from the Vcc of your Arduino, additional hardware (level shifters...) is required, so that you may be better off with a module of the right supply voltage. Or you use an Arduino board that operates at the same voltage, like a Pro Mini can work from either 5V or 3.3V. Consequently you may find modules or shields for either or both supply voltages, designed for use with an Arduino in general. This applies to the forum store or Adafruit products as well.
While a relay board does not require a library, it's sufficient to pull the signal inputs HIGH or LOW, a TFT or touch screen is somewhat useless without a library. Also all bus driven modules, using I2C, SPI or serial I/O, should be accompanied by according libraries, else you'll have to figure out their communication protocol (command and data sequences) yourself. OTOH many similar boards are based on the same controller chip, so that any library for that controller may be usable. In such cases it's extremely helpful when the controller type is specified in the data sheet. But unfortunately library documentation often doesn't include the used Arduino resources (timers, interrupt vectors...), so that some libraries can not always be used together in the same sketch.
Next comes the current drawn by a module, and a possibly required additional voltage for driving high current or voltage loads. Relay or motor driver boards typically require a separate power supply, even if their data lines have Arduino compatible signal levels. Other (low power) modules can be powered directly from the Arduino Vcc.
For shields a list of the used pins is important, to find out whether a pin is used by multiple shields, for different purposes. Bus signals, like e.g. SDA/SCL or MISO/MOSI, can be shared by multiple modules. The pinout of a shield also allows to find compatible libraries, which use or can be configured to use just these pins.