However, I often need to connect three wires together, so what I've been ending up doing is squashing two wires into one side of the connector and one into the otherside. But this doesn't seem like the best way to do it - what should I be doing here?
You can get Terminal Blocks - Barrier Blocks that have screw terminals that allow 3 wires on one side, and then use a jumper to connect the 3 on the other side to connect to 1 wire
This one has 4 screws, shorted across the barrier, for example.
You can also get jumpers to connect screws together.
Here's one example
Putting multiple wires into one side of a connector is what everybody does.
It does mean that it can be a bit tricky to get all the wires pushed in at the same time to the right depth.
With stranded wires, there are rarely any problems. Twisting them together before pushing them in can help with alignment.
With solid wires, there can be configurations of 'a few' wires that leave one of the wires not properly clamped, if the others form a stable arrangement and take the screw clamping force like an arch. Be aware of that as a failure mechanism to look out for.
If you really want a one-wire-per-screw arrangement, then you could permanently wire a link along the 'backs' of several connectors, and use one wire each to the front.
I suppose the question is subjective to what the best is, but imho the
accepted previously accepted answer is ridiculously expensive.
The latest accepted answer, screwing terminals, screwing wires under screw heads I've always found tedious to use. And don't see much of a difference between those screw terminals and the screw terminals in the OP question
(I know my opinion belongs in a comment but I don't have the points)
The simplest thus I consider best answer is simple twist on Wire Nuts. They come in all different sizes and are the cheapest and arguably the quickest solution for a "temporary setup"
But this doesn't seem like the best way to do it - what should I be doing here?
If these fancy terminal blocks are too expensive, which they often are, then normal screw ones can be used as well. With twin ferrules for example.
Three in one clamp is generally avoided. But for temporary setups it can work just fine.
You can probably get by with just twisting the wires together and stuffing it in the screw terminal. If you want to do it properly: use ferrules. You can twist the two wires together and put a twin wire ferrules over it, then place it in the screw terminal. Do the same with the single wire and a regular ferrule on the other side of the screw terminal.
With the type of connector from the picture, you can usualy put two (more if the wires are small gauge and the connector is large enough) solid wires into the same hole. You must twist them together. Ferules are useful only on multi strand wires. Do not use them on solid monolithic wires. It's good put some solder on wires twisted together. Waggo types are good only if all the wires have the same gauge. Otherwise there is a risk the smallest one won't hold.