I feel that I am suddenly seeing lots of apparent synonyms...how is κοιτάζω different from βλέπω?
Well, they are quite close. Still κοιτάζω is more like "have a look" and it is more suitable here, while βλέπω is closer to "I can see" or "I watch" and is something that we do for a little bit longer.
So I'm guessing you would say "Βλέπω την τηλεόραση" for "I am watching TV", and saying "Κοιτάζω την τηλεόραση" would be more like "I glance at the TV". Is this correct?
Βλέπω τηλεόραση (without την) = I am watching TV
Βλέπω την τηλεόραση = I see the TV
Κοιτάζω την τηλεόραση = I am looking / staring at the TV
In English, the verb is "look at" (the "at" is required). The phrase "I look the menu" is not grammatical.