I'll try to give a fairly generic answer to this broad question...
Avoid Dangers
This is obvious really, but first and foremost you'll want to avoid any kind of routes which lead you close to dangers. What's dangerous and what isn't and which levels of danger you'll be prepared to accept will depend highly on your skill level, your environments/weather, the mountain in question, etc.
- Avoid scrambles/climbs if possible (unless that is your actual goal obviously, in which case you should be properly equipped).
- Avoid passing steep ice/snow fields/glaciers. For many of those you need special skills and special equipment.
- Don't cross streams if avoidable.
- Try to avoid crossing ravines/mountainsides where rockfall is frequent. If you do have to cross do so quickly and alertly.
- Try to avoid spending time beneath cliff sides, as rockfall is always a danger and there is virtually no warning signs in such a case.
- Forests are generally quite safe to walk through (trees will protect you against rockfall, will protect the soil from eroding, etc.), but depending on the type of forest can be horribly uncomfortable and inefficient to pass.
- Avoid long wet grass, or more generally, anything slippery on long unbroken inclines.
- Generally walking on top of ridges will be fairly safe, unless the drop-off towards both sides is very steep and sudden. (In which case it can become a dangerous climb or scramble.)
Efficiency
Different countries/regions have different practices when picking paths up steep inclines.
In my experience this depends a lot on whether the route was ever used economically (e.g. travel up to mountain pastures with your livestock, or cross a pass to bring goods to markedmarket) vs. purely 'recreational' routes. I find the former to use more serpentines, where as recreational routes can be very steep/hard ascends.
- Personally I find it more efficient to pick a zig-zag/serpentine path up an incline - unless it is not very steep. This will allow me to keep a decently fast pace when ascending without getting exhausted quickly.
- But of course serpentines are a tradeoff of time vs. exhaustion. If time is more valuable, then by all means, ascend in the straightest possible line.
- Don't lose altitude once gained. E.g. no descends and re-ascends on the way up if possible. If you need to move around obstacles big or small, try to keep your altitude by moving along the hillside.
- Avoid forests if you don't rely on them to keep you out of harms way. Picking a path through a forest can be quite arduous.
Fauna and Flora
As always when following leave no trace be aware of your environment and the ecosystem you are moving in.
- If you happen to stumble upon any nests/dens, feeding places or generally herds of animals of any kind give them a wide berth.
- Be mindful of where you pick your path, as the mountain flora can be especially fragile due to the short growth periods.
- Avoid walking on raised bogs, as they will take a lot of damage from your steps.