Old-style thumb shifters are found on older mountain bicycles and modern touring or beach cruiser style bikes. Thumb shifters have a knob that you push with your thumb to move the derailleurs and change gears. Unlike their modern counterparts, thumb shifters typically do not have barrel adjusters at the shifter, but are adjusted on the dials at the derailleurs.
Ride the bike and shift through the full range of gears, noting if there are missed shifts or any chain grinding on the front derailleur cage.
Go to the rear derailleur's back and find the point where the cable enters the device. Locate the black dial barrel and turn it a quarter-turn clockwise to tighten shifting. Turn it clockwise if there are missed shifts off the thumb shifters. Ride and shift through the gears again to test the adjustment. If the gear shifting is still not hitting all gears, make another quarter-turn clockwise adjustment. Continue until the shifter runs through all gears.
Go to the front derailleur and locate the two screws on the back of the device. Shift so the chain is on the large chain ring. Look from behind to see if the chain is hitting the derailleur cage sides. If so, turn the lower screw a quarter-turn counterclockwise.
Ride and test the shifter on the left side of the handlebars. Push it all the way out and check that the chain moves across the chain rings and there is no grinding against the cage. If there is grinding, turn the set screw another quarter-turn counterclockwise and test again.