This is a hobby that has been done for decades.
Chances are if your photographing at a relatively large airport, most of the spotting locations have been found and are widely publicised.
Rarely though the opportunity does still exist to find new locations to photograph from at such large airports. Smaller airports seem to be less documented and photographed from and thus your forced to use your imagination to find new spots.
I always seem to find new spots to photograph from an ease. My mind clicks into over drive about the possibilities of vantage points and opportunities to produce a photograph from an airport that no one has done before.
Play by the rules
Our hobby warrants interest from airport authorities. If your in a new spot where local airport authorities aren't used to seeing spotters this typically raises the alarm as suspicious behaviour.
If your asked to move on, do it. Do it without delay.
Respecting their authority usually plays out well for the spotting community in the long run. You're seen as co-operating with them and wanting what is in their best interests.
Stay outside the fence
A pretty self explanatory one but none the less, worth stating explicitly.
Stay outside the fence and do not tamper with the fence.
Use Google Maps
Do some reconnaissance on Google Maps to evaluate potential options. It will save you petrol and time.
If your trekking into thick grass and its snake season (where I live in Australia, these venomous creatures can kill in less than 30 minutes) dress appropriately.
Don't try and jump across a creek with a 2 metre gap unless your the national long champion. Play it safe and cross that option off your list.
If there's someone in the aviation community who has a good relationship with the local representative's at the airport, leveraging their relationship to communicate with them about using a new spot would be smart.
Some of the best spots need not be close to the airport either. Perhaps a certain SID or STAR offers a perspective to get aircraft banking.
Sit down in front of the computer and spend some time thinking about what you want. Research, plan, execute.
Remember - If in doubt, stay out!