I would get one of
these,
these and
these. (and you might consider one of
these). You might want to shop around, and beware of the long shipping times out of China
Later, when you have a bit more experience, you could start using
these. (You can program them from your Arduino nano board). And you can incorporate them inside various modules where you need some control logic. (for the technical amongst us -- you set them up to use the internal clock)
If they're socketed (or using some other tricks) you can reprogram them if you find that you want to change their behaviour or fix some bug.
You will need to run everything from 5V, bit since you're using LEDs, that's not a huge issue. The arduino can drive a LED directly, and as you become more skilled, you can get one arduino to control many, many more things.
Once you have something working on a solderless breadboard, you can transfer it to a PCB or veroboard, or even matrix board, but there is an additional translation step there. There are some boards made which are
already wired like breadboards. These allow you to transfer a working prototype from a breadboard to a more permanent construction with almost no changes (beware the power supply connections though).
If you really want screw terminals, look at
these. They won't fit onto a breadboard, but they have the correct spacing so they will fit into every other hole on a board with standard 0.1 inch spacing (like the boards above).
Sure, you can also use shields and other things like that, but they tend to make it harder to move away from them for a final construction, and a little too expensive for my tastes to consume one (often with only a small fraction used) for each project.