Depending on who you ask, fungi either evolved much earlier or at the same time as the first land plants.

We can tell by looking in the fossil record. There is some evidence that says there are fungal hypae (the long strands of fungi, not the big mushrooms) that are present as early as 800 million years ago. But because the fossil record is not perfect, important physical characters of fungi are not seen that early. By about 420 million years ago we have evidence of both land plants and fungi at the same time. So it is difficult to say which actually evolved first.

On a separate note, green algae (like some of the seaweed in the ocean) are thought to be similar to the ancestors of land plants because they both have chlorophyll and photosynthesis. So it can be said that algae evolved before land plants.

We don't know for sure. Plants evolved, we think, during the Ordovician period around 500-440 million years ago, since we think that we have found spores of mosses that old. The oldest fossils known of plants themselves are Silurian, 440-410 million years old. Fungi do not fossilize readily, and there are few fungal fossils.

The Ediacara biota (c. 570 million years old) have been suggested to have been fungi , but this is doubtful (I think that they're animals). As a result, we don't really know just how old fungi are.