Minecraft – the optimum pattern to place crops in Minecraft

You need a light level of 9 or higher for wheat to grow. Saplings will need a light level of about 10 specifically above them, so one torch (which provides 14 light) nearby will suffice. There is a small disagreement on the wiki for what light level trees need, but 10 is the safe number. Beyond that light does not affect the rate at which they grow.

For wheat to grow quickly, however, you need to place it on hydrated farmland blocks, which meet the following constraints (from the Minecraft wiki):

1. Water at the same level, or one block above the farmland block. Water one block below will not work.
2. Water up to four blocks away, including diagonals.
3. The blocks between the farmland block and the water make no difference.

Farmland blocks are made by tilling a dirt block with a hoe.

Other constraints are that wheat will only grow on farmland blocks and trees will only grow on dirt/grass blocks. Also, trees need at least 5 blocks of open air above them to grow at all, but leaving 14 blocks will allow almost all large trees to grow (source). Space around the sapling is not required but will result in more leaves and possibly wood, so I recommend leaving an open 5x5 area that is also 14 blocks tall and centered on the sapling in order to maximize each tree's potential wood output.

Here is a good example of an underground tree farm.

The best pattern of mushrooms to make to grow them optimally

1. Flat ground. Mushroom spawning is 50% likely to happen at the same height, or 25% likely higher or lower by one block. If it can't be spawned there, then the chance is wasted.
2. Sparse planting. Mushrooms spread as far as two blocks away, so make enough room for them.

The rate of growth is only affected by the above two factors.

Other conditions that affect growth

If you can plant it by hand somewhere, it can be spawned there. No other factors. I.e.

1. Only at empty space (air blocks) with light level <= 12;
2. Only on opaque blocks.

The rate of growth at this optimal layout

Well, I can't predict exactly. But it should always be much slower than other plants' (it seems 100 times slower but i'm not sure).