It might be easier to start with your original suggestion, which is to receive signals from wireless routers and such kinds of devices. The reason for this is that it is easier to make an antenna with narrow directivity at higher frequencies. However, this is still a very challenging project.
One way to build the receiver is to use an evaluation board for a Logarithmic Detector IC. These ICs are made by companies like Analog Devices and Linear Technology. An evaluation board is better than starting with the bare IC because it is too difficult for the beginner to get the IC to work on his own. Many log detectors are broadband, so for receiving signals from specific devices you would have to put a band pass filter between the antenna and the log detector, a filter that is chosen for the devices you want to receive. For example, most WiFi devices operate in the 2.4 GHz band, so a band pass filter at 2.4 GHz would be appropriate for these. Cell phones operate in several bands including 1750 to 2000 Mhz, 825 to 960 Mhz for examples. Filters are available for all these bands.
It is possible to buy band pass filters in the form of a small circuit board with a surface mount filter soldered on the board and two RF connectors.
In order to build an antenna with a narrow beamwidth, such as you imagine, you will probably have to make something special. To start with, there are many antenna designs described on the web for 2.4 GHz Can antennas and this would be a good starting point.