No, sediments do not sink into the earth because they are too light. They stay on the surface and over very long times they form the continents. But they can get buried under many km of other sediments as parts of the crust they're on subsides or is pushed over other parts. Thus they can get into pressure/temperatue conditions where diagenesis (see other answer) or metamorphism alter them.
On a very general level, plate tectonics keep the crust in motion. It can cause the surface to rise in some places and subside in others. When it rises it forms mountains, for example in an orogeny over a subduction zone or when two continents collide and the crust between them is forced upwards. Weathering and erosion starts in the rising zones, and they produce a lot of sediments which are transported by water, wind and gravity down into lower areas. Otoh, where the crust is stretched or subsides, e.g. in Graben or basins, the sediments accumulate, layer by layer, and tell the long story of what has happened in the past (see @Universal_Learner's answer).
Another, somewhat more exotic process that forms thick sediment packages, is the scraping off of marine sediments from a subducting plate to form an accretionary wedge.
These sediment packages can get many kilometers thick, the Pannonian basin holds the sediments of a mountain range in thicknesses between 1 and 5 km, >6km in the Molasse basin north of the alps, up to 16km in the Upper Rhine Graben, where they can be altered metamoprhically (apart from contact or hydrothermal metamorphosis in regions with volcanism). Metamoprhic conditions ("facies") can also be met in a continent-continent collision and the connected mountain ranges, like parts of the Himalaya.
Rivers (i am not an expert in this and kindly ask for correction) do not rise above the surrounding (gravity is against it ;-)), but they change shape and appearance with the slope and sediment load. For example from "alluvial fans" at the foot of a mountain over "braided rivers" in mountainous areas where slope is still high and grain size big over meander patterns in low slope areas with small grain size to deltas filled with mud and clay.
When a river reaches an area where it can't carry on because all around the land rises again, it can build a lake until it finds a new exit, or it simply fills up the area with its sediments.