Use your Amiga with a modern flat screen
The first generation of Amiga computers (models 1000, 500 and 2000) date from a time before PC's had any serious graphics capability to speak of, let alone colour graphics. The now-ubiquitous VGA was mostly unheard of or ridiculously expensive, so Amiga was built with its own type of display connector. You'll be dealing with a 23-pin D-sub plug that no half-decent computer monitor will hook up with nowadays. You have a number of options, though.
On the cheap: Amiga to SCART
Many European television sets include a SCART connector. This interface with its bulky plugs supports a number of different types of video signals over a single cable. One of these is RGB mode, which is the poor man's version of component video. Discrete signals for red, green and blue are sent across the cable to be recombined inside the TV.
Amiga's monitor port just so happens to have pins which do exactly that: it sends out discrete signals for red, green and blue colour channels. Hopeful, but unfortunately these signals are incompatible with what a TV expects because they're much too strong.
Yes, you read that correctly. Amiga sends out too strong a signal for a SCART-enabled TV to handle. This means you may actually fry your TV if you hook it up to an Amiga the wrong way so this is indeed a warning.
For the stout of heart here's a schematic which seems to work with most TV's. It requires some rudimentary skill with a soldering iron, a cable, a SCART plug and a difficult to obtain 23 pin D-sub connector. This last bit was the main reason for me to chicken out and order a pre-built cable online.
Sadly this pre-built contraption didn't actually function on my TV, but it at least the connectors fit on both ends. The picture was 100% white on my screen, so I figured I had a lucky break and didn't kill my TV outright. After adding all the resistors shown in the schematic linked above, I actually got a very stable and sharp picture but it was also very dull. After removing one of the resistors, the picture became clear and crisp. Your mileage may vary, though, and do check the wiring of the cable yourself. Not all SCART cables are alike, and you do not want to feed the Amiga's 12V pin straight into your unsuspecting TV!
The elite solution: Indivision ECS
If you don't want to use SCART or intend to use an actual VGA monitor with your Amiga, you're going to need more than a length of wire and some cheap resistors. Sure, VGA is RGB like everything else but you'll need to tweak a few more signal properties for it to work. Without getting into the nitty gritty of things: the Amiga's standard output signal's refresh frequency is roughly half of the absolute minimum any VGA monitor will take.
There are quite a few products available on the market to upgrade the Amiga's standard video output to something a VGA monitor will accept. Most of these appear to be external and plug into the Amiga's video port. I intend to keep my Amiga 500 as self-contained as possible, so external plugs are out of the question. However, Individual Computers markets a product called Indivision ECS which does exactly what I need. Even though it costs EUR 100,- to buy I don't regret my purchase.
Installing the Indivision ECS means popping open your Amiga, wrenching the graphics chip (Denise) from her socket, plugging Denise into the Indivision ECS' own empty socket and plugging this combination onto your Amiga's empty Denise socket. I must say this is not for the faint of heart.
Extracting Denise is extremely delicate work. It is in fact really easy to break off pins, which may result in a problem that cannot be fixed. I had this happen to my Indivision ECS circuit board, but fortunately I was able to connect the broken pin to the Amiga's main board with a piece of wire. Don't count on this, though. Just be very careful when handling upgrades that go into chip sockets.
The end result is exactly as should be expected: clean, crisp and flicker-free pictures on any standard VGA monitor or VGA-equipped TV/projector I've thrown at it so far. My only gripe with the Indivision ECS is the short length of internal VGA cable. It won't reach all the way to the point in my Amiga's case where I had intended to make a hole for it. So for now it's just hanging out through the upgrade slot on the side of the Amiga.
If you only use your Amiga for games, get a cheap LCD TV and a SCART cable. This gives you support for all the video modes you need and it'll sort out audio for you in the same go as well. Picture quality is excellent with SCART, so there's simply no reason to bother with an A520 unless all you have as an ancient CRT TV with nothing but a traditional antenna connection.
If you intend to use your Amiga with a real computer monitor, look into the Indivision ECS. It may be an expensive upgrade, but it'll offer you a few extra screen modes if your Amiga is up to it. This hinges mainly on the amount of so-called chip RAM inside your computer.
To make full use of an Indivision ECS's extra screen modes, your Amiga needs 2MB of chip RAM but no standard first-generation Amiga has this by default. Upgrading chip RAM beyond the maximum originally intended by Commodore opens up a whole new can of worms and is highly dependent on the model and revision of your Amiga's main board. I may get into this at some later date, but for for the time being I'll be focusing on different parts of the machine to upgrade.