I assume, that your problem is how to connect everything to the power supply.
You have some different options, depending on your requirements. The linked adaptor (barrel plug to alligator clips) doesn't seem to fit the barrel plug of the power supply. So you cannot use that. Especially it seems, that both are the male part of the connection. For every plug connection you need a male and a female part of the same connector type (much like your power outlet on the wall is a female part, while the power plug on your supply is a male part; in terms of plug systems).
The best way (especially, when you want to not only experiment, but also use it as a fixed product) would be to solder the wires. For this you need a soldering iron, soldering tin and something to clean the iron (sponge for soldering irons for example).
- You can buy a barrel jack, solder wires to it and connect them to the Arduinos 5V pin, ground and to the LED strips power lines in parallel (from + pin of the supply one wire each to 5V Arduino and + pin of the LED strip, from - pin of the supply one wire each to ground of Arduino and - pin of the LED strip). Then you can still disconnect the power supply from your project.
- If you don't want to buy a barrel jack and you don't need to disconnect the power supply from your project, you can splice up the cable of the power supply (only the LOW VOLTAGE PART!!!). You can use nail scissors, if you don't have other tools for it. Cut off the plug by cutting the cable a few cm before it. Then cut the plastic between the 2 connected parts of the cable for a few cm, so that you get the 2 individual cables (+ and -). Use the scissors to carefully cut the outer plastic around the cables and remove about 1cm of it, so that you now have 2 cable ends with the bare cable core free. Twist each end, so that it each little copper strand stays in the cable shape. Then solder the cable ends. Now you can take another wire/cable to solder the connections from the two ends to the described parts. Be sure to isolate the connections at the cable (in the best case with shrinking pipes, otherwise you can also use tape).
If you don't want to solder, you can search for fitting connectors for every part. There are breadboard power adaptors, which should work with this power supply. They have pins on them, that you can not only use with breadboards, but you can also connect jumper wires to them. There also exist such adaptors for typical LED stripes, though that depends on the type of strip, that you have. When looking for jumper wires, you should make sure, that they are capable of conducting enough current. The Arduino does not need much current, thus standard jumper wires will be sufficient. But the LEDs can draw much more current, so that the standard jumper wires will start to smolder and catch on fire. Then you need some with a bigger wire diameter.
You can also use alligator clips, but in my experience, they only work for rather specific cases. The connection, that they make, is fragile and can be easily disturbed. They are slipping and rotating easily. When connecting the LED strip, you might easily connect + and - together and destroy the power supply. Also it is almost impossible with most of them to connect to a standard 0.1 inch pin (which are used on the Uno and Nano), without also connecting other pins. Unless you break out these pins with something else (like jumper cables), that is not a good choice.
You can now choose yourself, which way to go. If you plan to do more with Arduino and electronics, I would suggest, that you buy a soldering iron and the needed supplies. You will need it and they only cost a few bucks, if you just buy a simple one (back then I got one for 15€/$, which also had a temperature control, so you can also get it cheaper). If you don't want to buy it, but still want to solder, you could look for a maker space near your location.
I would go for soldering here, because you are very much limiting yourself, when you choose to do not. Then you need to buy fitting parts, which is more difficult.
how do I know what the ground is and the 5v?
For the power supply you can see it on the label. There you will find the following sign:
It means, that the outer shell of the connector is ground (
-) and the inner pin is 5V (
+). With cables there are 2 possibilities:
Often the wires inside a cable each have an own isolating plastic shell around them, which is color coded. Black is normally ground, while red is 5V/+.
When having a 2 wire cable, the cable is often designed as coaxial cable (which reduces noise on the signal). In that case you have one inner wire, which has it's own isolating plastic shell, and the outer wire, which is laid around the inner wire without any further isolation. The outer wire is normally ground.
When using an extra cable for connecting 2 parts, make sure to use the same wires on both sides (for example the red wire for 5V on both sides).
To be sure, you can check the voltage on the wires with a multimeter (also a very useful tool), when the cable is connected. A multimeter has a positive and a negative lead, which are marked as red and black respectively. When you connect the multimeter to the wires and it shows a positive voltage, you have found the correct association of the wires with their voltage (5V or ground).