The largest canyon, the Valles Marineres is so deep is a "rift valley" like the Grand Canyon or the African Rift Valley. The Mars crust was pulling apart - on its way towards splitting into continents. But it never managed to get very far, so all we have left of its phase of attempting to form continents is the Valles Marineres.
Valles Marineris in close to true colour and four times height exaggeration.
Fly through: That's also why its volcanoes are so large. Without continental drift they just stay in one spot and get larger and larger.
Most other river like features on Mars are results of floods, and seas in the ancient Mars, some are channels caused by huge floods. Some of those caused by meteorite impacts heating up ice .
Smaller features show effects of dry ice and wind and dust. And some of thoee are still forming today. Including a few rare cases where liquid water may be a part of the story, even today, in thin film type flows most likely. And if there was a massive impact say by a comet (if Sliding Hills had hit Mars) you'd get temporary lakes and floods and perhaps gullies forming and rivers, just in the region around the impact crater. They would quickly freeze over or evaporate.
But after a big impact even in modern times, say a comet of 1 or 2 kms across or larger, which has to happen from time to time - if it impacted into icy regions in the upper lattitudes, you might have an ice covered lake and a newly created hydrothermal system of hot rocks below the crater which could keep the lake liquid for up to several thousand years (in the theoretical models of it anyway).