Rear derailleur does not shift onto sprockets 2 and 6
This sounds like a classic case of the shift cable hanging up in the housing. As the shifter releases cable increments to shift to smaller sprockets, the derailleur spring cannot pull the cable though the housing properly.
You can try removing, cleaning and lubricating  the cable. Replacing the cable and housing is the proper solution though, and is relatively inexpensive.
While you have the cable detached, check the derailleur itself is moving freely.