My cousin replaced a instrument panel cluster of a car with another one that is very similar. He did it by cutting the original junction/wires block that come out of the instrument panel and attached the junction block/wires of the new cluster to the wires by twisting them together and insulated with insulating electrical tape.
Would it be better to solder them and insulate?
While the twist/tape method will make the connections and work, I tend to leave this for the quick fix. Over time, with the everyday vibrations occurring, this method tends to allow for continuity issues and things stop functioning.
Another easy type fix is wire nuts. This takes very little time, is more secure than the twist/tape method, and doesn't require any special tools. Like the twist/tape method, wire nuts can loosen up over time and come lose, but would usually last longer than the twist/tape method, plus while the nut is in place there's virtually no chance of a stray wire poking out and causing grounding issues. (Added from a comment by Nate Eldredge ... thanks for the add!)
An intermediate type fix is to use crimped butt connectors. When done correctly, this method has great longevity, but may have issues down the line. They are not impervious to vibrations or poor application. This requires the use of crimpers and the butt connectors themselves, so take a little more time than the twist/tape method. Also, the quality of the butt connectors has a large bearing on the finished product.
The far better method for a permanent fix is to solder and heat shrink application. This takes a little bit of skill, some additional tools (soldering iron), and some time. When done correctly, the connections are usually permanent and there aren't any further issues with them.
All-in-all, there's no wrong way to do it as long as there's a connection and what you are fixing works in the end. I'd suggest there are better ways of doing it, which I think I reflected in my write up.
@Pᴀᴜʟsᴛᴇʀ2's answer is excellent. But, I'd like to add: rather than just twist & tape, there's something called a "lineman's splice," or "Western Union splice" which provides a great electrical connection as well mechanical strength, and is easy to do. (see Wikipedia's illustrations., or google "lineman splice" or "Western Union" splice - there are a few good videos on Youtube.)
The technique gets its name from its use in repairing telegraph and phone wires. If it was strong enough for wires out in the elements, strung between poles over many miles, it's probably strong enough for your instrument cluster.
In my experience, this holds up better than twist caps, and you won't wind up with the situation where you've twisted the cap on, and the wire breaks or one of them is miraculously not secured inside the cap. And, this doesn't require any special hardware.
This is more common with solid-core wire than stranded, but it works just as well for stranded wire.
( from Wikipedia)
The idea is:
Shouldn't take more than about 10 minutes per wire. If done properly, it'll last longer than the car.
If you don't have heat shrink lying around, you can get it from your local auto parts store for cheap. You could use electrical tape, but I despise electrical tape. It's sticky, melts easy, and just screams "amateur hour."
Another frequently used alternative, cheap and which requires no special tools, is the screw terminal.
There are various types, perhaps the most common is the two-way one:
or the one-way alternative:
In the latter, both wires enter the same end twisted and are secured by the screw.
