Is there a certain time of year that is best for planting new trees to give them the best chance of survival?
The answer is....it depends on the species, the kind of weather and your location in the world.
Oak trees are usually planted in the spring in the Northern Hemisphere. The city where I live commonly plants maples, gingko and hackberries in the fall, sometimes into October.
If you are in a drought it's not a good time to plant a tree no matter what the season or species. If you must plant in a drought, be prepared to water, a lot. The first few seasons are critical for the development of a good root system and and structure for a tree.
To provide a counter example if you are planting willows in a wet area they can be so weedy you could do it almost any time you can dig the ground.
Bare root stock is almost exclusively planted in the spring with balled and burlapped material which is normally larger being planted spring through fall.
Evergreens in colder areas have a greater chance of success if they are planted in the spring so they have the whole summer to root. Winter's drying winds can burn an autumn planted evergreen. The damage to the foliage can be seen for years with some species.
I'm sorry if this answer seems a little scattered but you have asked a wide ranging question so you get an answer going in a lot of directions.
Edit: @bamboo Yes, the differences in planting evergreens are due to the much lower winter temperatures that the northerly States and provinces receive. The last few years have also seen much more variability in weather which is harder on evergreens.
I don't know where you are in the world, nor what trees you're thinking of planting, but the general rule is simple - autumn is the best time, or early winter if they are bare root. This should mean they will have a plentiful supply of water over winter, enabling the roots to settle in ready for the following season. They will, though, need to be given water at regular intervals (a good soak) during dry spells the following spring and summer.