I'm studying up on EE and I have a budding grasp of circuits and ohms law so far. As I continue I wonder how feasible would it be for me to read data from existing electronic devices such as this temperature/humidity gauge. My end goal would be to wire this into my arduino. I realize there are existing plug ins for the arduino for this purpose but having the knowledge to do this with my existing devices seems very exciting. So are manufactured electronics like this hard to wire into? What would that involve?
The question is very open ended as it really does depend on the individual thing you are trying to hack into. Sometimes they have every chip in a nice big DIP package with all the markings plain to see, with the pcb nicely labeled. Other times you get Chip-On-Blob packages with nothing labeled, no accessible test points, no markings, smd chips so small that you can't easily connect a wire to it or bga packages where the pins are underneath the chip body.
The easiest thing to do is open it up, and take a look. Google any part numbers you see, look for projects that access them, look for datasheets. That's the real starting point.
Then there is the question on if you are simply trying to repurpose the item (easier), or if you are trying to read data without affecting how it works (harder). Sometimes you go about it a different way, can't access the raw chip data, how about accessing the lcd's memory and figuring out what is being displayed. There are multiple ways, but you need to know what it has first to be able to come up with them.
For a relatively simple gizmo like the gauge in the O.P., the from-scratch approach is more feasible. You can get individual components (such as sensors, displays) by themselves or as Arduino shields. You'll have fewer black boxes to deal with. Industrial design is just about the only advantage one can get from hacking/hijacking a gadget like that gauge.
Exception: There are gadgets which have already been hacked, and proceedings have been posted on the web. Subscribe to a site like hackaday.com, and eventually you might come across something that looks appealing and doable for you.