If you mess up, don't worry! It's pretty easy to re-melt solder and redo your connections.
Most of the time, you can just re-melt the solder with your soldering iron and pull the wire off, trim it a bit, and try again.
If you've just got way too much solder on your joints and the pads are melting into each other, reach for your solder wick. This stuff is made of braided copper and solder absolutely loves it. Get it in between your soldering iron and the solder pads, and it will soak up any extra solder that's in your way.
The Sacrificial Pixel
Sometimes you're soldering to side-light pixels or 144/m pixels where the pads are teeny tiny. How the heck are you supposed to securely solder three whole wires to those tiny little pads when they've been cut in half? Sheesh.
The trick here is to use what's lovingly called the "Sacrificial Pixel" method. Instead of cutting through the middle of the pad when you trim down your strip, you can cut one pixel out entirely and leave full pads on either side. This pixel becomes useless, since the pads are cut off on both sides, but the strips remain functional and much easier to work with.
I do this pretty often, even with the strips that have larger pads. I like keeping my sanity intact (more-or-less), and this helps. Just remember to account for the missing pixel in your planning, if your project needs to be an exact length.
The Insulation Gap
Silicone stranded wires are absolutely my favorite. One reason is because the silicone coating is so very stretchy and flexible that I can grab it with my fingernails and pull, and expose a bit of un-shielded wire without cutting the wire at all. This is difficult or impossible to do using cheaper wire with regular plastic insulation.
This can be so helpful when there's not much play in your wire lengths but you need to split your wires for some reason -- maybe you need an extra ground connection or you want to splice on another LED strip. It's really useful when making repairs on an installation as well.
You can't get heat shrink on using this method, so be sure to seal up these connections using hot glue afterward.
One more tip: if you split the insulation and then bend the wire 180 degrees, you get a little exposed nubbin of wire that's neat and tidy with no fuzzies. You can solder this onto your NeoPixel strip an get a double-wire connection to your solder pads.
This is great when you have a larger number of LEDs and need to power them directly instead of through the microcontroller, or if you want to create an easy DIY power bus between strips.
Solder Seal Connectors
Sometimes there are places you can't reach with a soldering iron. If you're making wire connections inside a larger art installation, or doing repairs on a costume while you're at a festival, it's good to have a solution you can use in the field.
These solder seal connectors are really cool. They're basically heat shrink tubing with superpowers. The two colored bands are filled with hot-melt glue, and the center silver band has low temp solder inside it. All you need to do is slip these on, make your wire connection, slide the solder seal connector over it and hit it with a heat gun or lighter to shrink the plastic and melt the goodies inside.
These things are lifesavers. They aren't in the Adafruit catalog at the moment, but here's a link to order from Digikey.
Sometimes you have a spot that just keeps on breaking. Waistlines or elbow joints in costumes seem to be particularly vulnerable -- you keep repairing it but it just keeps breaking.
If you're having this issue, one workaround is to connect the two halves of your strip with a couple inches of wire instead of trying to reattach the strip directly. This silicone stranded wire is really flexible and will tuck out of the way easily. You may get a slight dark spot at the flex joint, but it's much better than a costume that just keeps breaking.