how to tighten the chain on a ten speed bike?

the easiest thing to do is replace it. The chain is most likely stretched, a new one is 10–25 bucks and will make a world of difference

The chain on a ten speed bike must maintain slack so that it can move on the gears as you move the shifter. The tension is supplied by the rear derailer and its springs. If it is not shifting correctly it might just be out of alignment. First, oil the chain and the gears to make sure everything is moving smoothly. There is a adjustment in the rear where the shift cable goes in. It will turn left or right to adjust the tension on the chain. You can turn this one or two turns (keep a count!!) in one direction to see if the shifting improves. If not, go in the other direction, but remember to count the turns so you can return it to it’s original position if you need to. It will not take many turns to have an affect, so go slow. Turn the bike upside-down (on the seat and handle bars) then turn your peddles and shift the rear gears to see if it is shifting better. Try both directions, but do not turn a lot of turns. If this does not help, then take the advice of “Hawaiiguy” and have it replaced. Take it to a good bike shop. They do this in their sleep.

I will second what woolval said.

Turning the adjuster barrel is a good idea, just be sure not to mess around with the limit screws on the rear derailleur – if you adjust those there’s a chance you might shift your chain right into your rear wheel (bad). Is the shifting bad? Why do you want to tighten the chain?