The front derailuer on my daughter's bike won't allow it to shift properly. The gears are Shimano Tourney with a Grip Shift system. If I could get the set instructions on how to align it properly I think this portion of the bike was put together wrong by the store that I bought it from.

Main thing with gears is that they have the same number of gears, and that the cage is the same length. Also the RD TY15 has a built in hanger.

The Tourney pictured has a built in hanger (below link) just make sure the replacement does.

SHIMANO TOURNEY TX 55 Rear Derailleur with Hanger Virtue Bike

This may be to do with the replacement having a shorter cage, so it can't go on the big sprocket and big chain ring at the same time.

Try to not use the big rear sproket and the the big front chaining at at the same time,

Put the bike in the lowest numbered gear, put chain on smallest spocket/chainring. Disconnect cable from derailuer, pull cable, then replace with a new one by following steps in reverse. For detailed pictures, go to Utube or sheltonbrown .com.

Maybe go through the whole procedure. Web search "adjust front derailleur" for videos.

http://www.parktool.com/blog/repair-help/categories/derailleur-systems

If you're saying the shifter itself isn't staying in the selected position and is always relaxing ALL cable tension, you have a defective shifter. Some, like the Grip Shift MRX, seem to rely on the handlebar grip to keep all the guts inside. I know of one where a space developed when the grip slipped outboard a little bit and the shifter parts separated enough for a key piece to fall out.

No one sells the nickel part so you'll probably end up replacing the whole shifter. You may even have to buy a complete set. If you have to go that far and it IS a defective design I'd get a Shimano shifter that works with 6-speeds. Like Revo. Look on eBay.

OR, maybe the cable is just really loose.

This resource is very useful...

http://www.parktool.com/blog/repair-help/categories/derailleur-systems

Derailleurs are mass-produced and are essentially identical to each other, so unless damaged it's unlikely to be the derailleur. A sticky cable could keep the derailleur from trimming properly. It also can lead you to think it's tight enough when it is not. Otherwise it's hard to diagnose remotely. Derailleur must be both at proper angle and height to shift well, cable both move freely and be at proper "tension."
Di2 Shimano Electronic Intelligent System Installation

I’m happy to help further over the phone at https://www.6ya.com/expert/donald_f2ed37026a3ac881

There could be several reasons for the shifting to not be working properly. The most common is that there is too much slack in the cable and you need to re-tension the cable. There could also be something broken in the shifting system or bent.

Best thing to do since index shifting requires precise fixing to make work properly is too take into your local bike shop, they can diagnose the issue very quickly and let you know what is the remedy.

first identify the problen is it front or rear derailuer.secondly bike shop mechanic hate cheap mass produced bikes because they do not sell them
so lets solve your problem,okay sometimes this can be an adjustment issue.now if its bunching on front then first read your owners manual to see which gears are compatible,sometimes it will suggest not using certain gears,secondly if yhe chain has fallen off and got jammed it may be bent and need replaced visually confirm if the chain is straight.okay if it is not the chain then the length and length adjustment can be changed using the adjustment on the rear derailuer look and you will see 3 screws located on it the two with L /H are for adjusting the gear stops l is for low gears,so it does not go too far and wedge between the gear and spokes,H is for the small gear(high)if it goes past and falls off adjust this one.lastly the one located at the very rear adjusts chain tension the loosening of this one will relax the chain and apply more chain on the gears   tightening of this one will tighten the chain,adjusted properly your chain should not bow down over adjustment will cause chain skips.now if there is too much slack in the cable that is adjusted by the adjusters on the ends of the cable where it meets the shifters,learning to adjust your bike will save you a lot of money in the long run.
hope this helps   

Count the number of sprockets you have up front, then count the number you have in the rear, Multiply, 3 x 6 = 18 speeds. 3 up front, 6 in the rear, or 2 up front, 5 in the rear, 10 speed. Etc, they go as high as 21. You might only have one in the front.
To shift, you have to be peddling, get the bike going, put the chain on the middle or the biggest sprocket in the front, move the left thumb shifter to do this, do not let the chain rattle, make it a clean shift and don't let it klik or rub. Then put the chain on the biggest one in the rear , without any rubbing, chattering, slipping, etc,with the right shifter, now you are ready to take off easily, when you get going and the pedals get too easy and go faster, shift the rear to the next sprocket down, then again, then when you are coming to a stop, down shift the rear sprocket back to the biggest gear and you will be ready to take off again easily. A few tries, you will shifting like mario andretti. Hope this helps.

There are several things that could be wrong. You could have thrown your chain (you couldn't ride at all if so). The bike could have been crashed and the derailleur bent. The cable might have slipped where it bolts to the derailleur. Or the cable or shifter could need maintenance, too. It is most likely one of the above. If it is only the adjustment, there are two stop screws on the derailleur that can be adjusted.

My best advice for you is to either find a bicycle repair manual at a local bookstore or your library that can show you some of the detail. Or just let your local bike shop do the repair.