How to testfor and execute the effect in the single command block?

No, you cannot chain testforblock with more commands just like that, but there are other ways to do it.

1. execute's detect keyword

The execute command has a special detect keyword that allows you to do exactly what you want. It is essentially a testforblock query inside an execute.

execute <entity> <x> <y> <z> detect <x2> <y2> <z2> <block> <data> <command> 

This will run <command> on <entity> only if <block> (with <data>) is present at <x2> <y2> <z2>¹. So for your example, you want to use

execute @p ~ ~ ~ detect ~ ~-1 ~ grass 0 setblock ~ ~-1 ~ dirt

^{1) If you use tilde notation for relative coordinates, the second set of coordinates is relative to the first set (in this case the position of @p).}

2. fill's replace option

An alternate way to do what you want to do is to use the fill command with the replace option (This is more limited in scope, and I only include it here for completeness):

fill <x1> <y1> <z1> <x2> <y2> <z2> <TileName> <dataValue> replace [replaceTileName] [replaceDataValue] 

If both coordinate sets are identical, it will only affect a single block. So, to replace grass below the player with dirt, you can use

execute @p ~ ~ ~ fill ~ ~-1 ~ ~ ~-1 ~ dirt 0 replace grass 0