Those RF modules look relatively easy to apply. I note that the CC2500 is an integrated RF transceiver IC, but will be tricky to get working properly without a correct pcb layout to properly implement the output/antenna circuits and to provide low impedance bypasses. At 2.4 GHz you have to be careful to use well-designed transmission lines (like microstrip) and choose good quality SMD inductors and capacitors, preferably 0402 sizes. Such things are typically not easy for the beginner. That's why using an RF module like the radiotronix will be a lot easier and you are more likely to succeed at your ultimate goal which seems to be to get a signalling system to work.
If your goal is to learn RF, then a module won't get you very far because most of the work is done for you. But jumping into a 2.4GHz design for your first project is putting the cart before the horse. So go with the radiotronix modules if possible.
By the way, I would not use their antennas if I had little money. After all, it is very easy to make a quarter wave monopole antenna with a simple piece of wire correctly mounted on some metal. You could save yourself the $4 for the antennas if you don't care how they look.
In the meantime, I will look for an example RF transmitter and receiver design using very few transistors. I should have mentioned before, but the idea of using 1.6 MHz does have one major disadvantage. This is that it is hard to make a really good small antenna for this low frequency, so your range of communication will be quite limited and probably entirely dependent on how big an antenna you can fit. In fact, the best antenna may likely be a coil of wire of, say, about 20 mm diameter, mounted on your circuit board, instead of a wire sticking out of the box.