wow... that's a couple of questions that would take a long to fully answer!
For breadboarding/prototyping a circuit without soldering, there are things called solderless breadboards that have been around for decades and very handy when using leaded components. Adafruit is a business that sells a lot of stuff to hobbyists, and they sell a number of these.
https://www.adafruit.com/category/82
You can also get them from the mailorder electronics houses too, such as Digikey:
https://www.digikey.com/products/en?keywords=solderless breadboard
As far as soldering stuff to a board... that's another can of worms. There are lots of ways to do it. I'll be that Adafruit has some instructions, and youtube is bound to be another source. I'm guessing that Dave Jones at EEVBLOG has done videos on this, and Great Scott on youtube spends so much time building up circuits that he's probably done something about prototyping too.
Personally, I've been building small circuits with surface mount component (i.e. quite small stuff), and just hack up copper-clad circuit board with a dremel tool in order to make a circuit board. Not pretty, but seems to meet my modest needs.
Here's a little two-transistor LED flasher circuit that I built recently: