My two go-to glues for plastic are Instant bond and ABS cement
If the plastics are both ABS or I have a part that broke and is made of ABS plastic, the latter is the strongest bond you're going to get. It basically fuses the plastic together. When that doesn't work, I use the Instant bond glue (although I never use that activator that comes with it). The Instant bond I discovered by looking at MSDS sheets, is the exact same component as this stuff I was buying under the name of "The Last Glue". The difference is the Instant bond brand is thicker (and therefor much easier to work with) and the price is a fraction of what I was paying for the "The Last Glue". This is a significantly stronger bond than a typical Cyanoacrylate glue. Not all, but many things I glue with this never break in the same place or come apart again.
Instant bond is very different than a typical Cyanoacrylate adhesive (aka Krazy Glue). it bonds in the absence of oxygen, rather than by a rapid polymerization in the presence of moisture. That gives you a really long open time in most cases, until you're ready to bond and thus cuts off the air when the two pieces join. Supposedly it also makes it possible to roll your fingers apart if you accidentally bond them, but I've not found that to be the case. Bonds skin quite well! 
Acetone is the solvent to remove it from your skin or de-bond your fingers. That low VOC nail polish remover doesn't work, so if you're not going to wear nitrile gloves while you're using it, you might want to also pickup some Acetone while you're at Home Depot. You can make this glue last a really long time (around two years or more) by storing the bottle in the fridge.
Clean both parts with Isopropyl Alcohol before bonding for the best results. Also, when you're done with bonding, I found it keeps the end relatively clear by squezing the short side of the bottle to suck any beads of glue back into the bottle. Otherwise, even though there's a pin in the cap to keep the nozzle clear, the oxygen gets cut off in the gap between the cap and the nozzle when you close it and the glue hardens around the end of the nozzle, making it more difficult to use the next time.
It's counter-intuitive to use at first, because you're always trying to keep the air (and thus the moisture) from getting to Krazy Glue and similar, so you're always sealing the tube right away. But this glue doesn't bond when in open air, so you can keep the lid off the bottle while you're working and it won't hurt a thing. You need so little of it, I often put a drop on wax paper and then touch the piece to the glue, rather than applying the glue to the piece from the bottle like you have to do with traditional Cyanoacrylate adhesives. The drop will sit there for hours on end and you can keep dipping parts in it as long as there's enough glue there. It won't harden sitting in open air.
