Climatology is extremely complex, and there are pieces of the Earth's climate which scientists still don't fully understand. Every moving piece interacts with everything else, so any small change can have huge and far-reaching effects.
Here again I must warn you that my understanding of climate science is hugely inadequate. I am an amateur, and I am attempting to provide a "good-enough" level of understanding for other amateurs trying to make reasonably believable worlds. No one is going to go study the climate of your fictional world, so you really only need enough information to plan reasonable climate zones.
The rotation of a planet and the position of mountain ranges are hugely important to your world's climate. Be sure that you have you planetary geography finalized before you move on.
Because the atmosphere isn't physically attached to the planet, the air around the planet moves unevenly. This is called the Coriolis Effect, and it's very complicated. Take a look at the third graphic down from that wikipedia link (the one with the globe and circular arrows): That's the important part. Notice that the circles get bigger near the equator, and that they are clockwise above the equator, but counter-clockwise below the equator.
Wind can also have a dramatic effect on a region's temperature range. The "Jet Stream" carries warm air up to western Europe, much of which would normally be much colder than it is.
Water follows the Coriolis effect too. The Atlantic and Pacific oceans have huge currents which wind in rough circles in the northern and southern hemispheres. These currents distribute cold and warm water, and carry all manner of aquatic life all over the world.
Water and the water cycle are two of the most important pieces of your world's climate. Rainwater hydrates huge parts of the world which aren't permanently irrigated by rivers, and the lack of rain defines arid areas like deserts. Regions near rivers may be abnormally verdant for their climate, like the banks of the Nile river.
It's also important to consider the effects of water on the ground itself. Water picks up bits of soil, stone, or whatever it runs over and carries them downstream. This allows rivers to gradually carve permanent beds into the earth. Larger rivers have more profound effects, and can carve huge canyons (like the Grand Canyon) into even the strongest of stone over the course of millenia.
Floods can also have interesting effects, as the drying water deposits whatever silt the water had previously collected onto the surrounding ground. Regions which flood frequently (usually during the "rainy" season, or in the Spring, depending on the local climate) often form "flood plains", and have extremely fertile soil, allowing very effective farming between floods.
Temperature is a huge part of climate. You planet is generally warmest along its equator because it is the closest to the planet's star on average. The Earth tilts relative to the sun during its year-long seasonal cycle, creating alternating summer/winter cycles. Regions near the equator tend to be very warm, and regions near the poles tend to be very cold. Planet's poles often have permanent ice caps which grow and shrink seasonally. These ice caps might temporarily connect continents which are typically separated by water during the summer.
Actual fires also have an important role in the climate of many parts of the world. Southern California has an annual fire season which, while devastating for the human population (and often for local flaura and fauna), is an important and long-standing factor in the local climate.
I am captain imaginary planet!
Example 1 - Shadow of Olympus
While we can draw a lot from real-world Greece's normal temperate climate, we need to deviate a bit to make some significant changes to account for our unique geography, and the magical forces at play around the world's edges. Places near the plane of water will be cold and wet, places near the plane of fire will be hot and arid, and places near the plane of air will be exceptionally windy. Places near the plan of earth will most strongly resemble normal, real-world climate.
Example 2 - Space Grease
Climate doesn't matter much... in space!
Example 3 - Heroes of Tonesvale
Tonesvale should have a good mix of weather to allow for lots of interesting plot devices, so let's go for a reasonably temperate climate. The local mountains allow for plenty of rain, making the local soil very fertile. Reasonable hot and cold extremes make it a pleasant place to live with occasional snow in the winter and heat waves in the summer.