Strictly speaking, the only elements required for a city are:
- Power, since zones do not grow without electricity
- Residential and industrial zones, to provide homes and jobs for Sims. Commercial zones are not strictly necessary I believe
- A fire station, since fires are disasters which pauses the game and will destroy an entire city if left unchecked
And... that's about it. A city which contains only the elements above will be a fairly crappy city, but it should survive assuming you can finesse the budget enough, and live with how poor it is. SimCity 4 is a sandbox game - most elements of the game are by definition unnecessary. Now, about elements of the game which which may not be very good value for money
Agriculture is good as a form of employment. Unfortunately they don't provide any tax revenue and creates terrible water pollution (easily avoidable by placing towers and pumps away from them). The only benefit they bring is allowing you to build the farmer's market reward structure
Landmarks are very useful as commercial demand boosters. Unfortunately they're usually too expensive for that purpose. I recommend the Functional Landmarks mod if you wish to build them.
Airports boosts commercial demand, but like landmarks smaller commuter airports aren't usually worth the expenses. Larger cities can still benefit from larger regional/international airports
In the end, the question as to whether something is worth building is always "it depends". Every choice you make have tradeoffs, and making the best of what you have (short of using mods) is part of the fun.
If you need more information, http://www.gamefaqs.com/pc/561176-simcity-4/faqs/21501 contains excruciatingly detailed information on every building and game concept. Well worth a read if you want to know if something is worth building.