Not necessarily, mount your harddisk in WinUAE and install it there then pop the harddisk back into the Amiga. :D

I also would recommend HD/CF/... with OS3.1 and/or 3.9. At least for using trackers, paint programs and gaming. An accelrator with enough ram would be fine.

A quick rundown of the amiga I think might be useful for you here.

Im over simplifying things here, but given what you've asked this should hopefully tell you what youre wanting to know:

Firstly, and a500 and a2000 are actually pretty similar. Both use whats known as the ocs (or ecs) chipset, which deals with graphics and sound. This chipset is 16bit and has a palette of 4096 colors, of which up to 32 colors can be displayed simultaneously (or64/4096 with special ehb and ham modes (both of which has special characteristics in regards to color use)). The standard cpu in these machines is a motorolla 68000 running at 7.14mhz.
The a2000 however also has expansion slots, allowing a person to add all sorts of upgrades, ranging from graphics cards (faster and more colors/higher resolutions, etc.), sound cards (for improved fidelity and bitrates vs. the inbuilt (Paula) audio), network cards, and a whole range of others. A harddrive is also more easily fitted than it is in an a500.
To run a game like Doom you'll need to upgrade the processor and ram beyond the default specifications of either machine, much the same as you wouldnt be able to run doom on a 7mhz pc with 1 meg ram. There's a variety of cpu/ram upgrade options available for both. Something like a 25mhz 68020 or 68030 cpu is probably bare minimum for Doom, inline with required specs on other platforms. There are a few more simplicstic clones that have lower required specs though.

An a1200 could be looked at as a 32bit replacement for the a500. It supports a 14.28mhz 68020 cpu by default and has the AGA chipset, which provides a moderately upgraded graphics display, allowing for up to 256 colors to be displayed simultaneously (or 262,144 colors with a special HAM8 mode), and at higher resolutions than the OCS chipset. Given that you'll be getting an a500 or a2000 this isnt overly important, but I thought I should mention it for sake of clarity.

In regards to the Operating system the 680x0 series of cpus can only run up to OS3.9, but for a machine with a 68000 cpu you're more or less restricted to OS3.1. While this may initially seem not favorable, bare in mind that there's no major differences between between even OS3.0 with assorted updates and enhancements, etc. applied vs OS3.9.

Both can infact be made to look and behave a lot more like the newer versions of amiga os (and its clones) than the default OS3.9 installation. I wont go into enhancing OS3.x here though as its a topic that needs its own thread.

So yes, an amiga 500 or amiga 2000 can do things like playing Doom, getting online, playing music, etc. but they'll need a few upgrades to do it in the way the youtube vids it sounds like youve been watching demonstrate.
Having said that though, given some time and effort I think you might be surprised what you can do on even a default specced a500/a2000.

Good summary there fishyfish :) One thing I'd like to mention though:

To be able to play Doom (we're not talking about the limitation to 32 colours without a suitable graphics card) at a somewhat playable speed, you'll need a 030 accelerator which will cost quite some money. 'Getting online' will require a TCP/IP stack software plus the current 68k browsers are quite limited regarding surfing the internet. I know that it might sound strange, but if you'd like to play Doom and surf the net... try to get a 'space debris' class PC for free and use that. It'll save you a lot of headaches and money in the long run :)

To each his own as they say, but Id guess the OP already has a pc/mac/phone/whatever capable of using the internet (and/or playing doom), so the fact that certain tasks might be better suited to devices other than the amiga I suspect isnt really of interest whereas doing these sorts of things on older machines like the Amiga is. Only the OP knows for sure of course.

Either way its not nearly as bad as if often suggested, and its not like the web is the entirety of the internet :)

Let me put my last post in a different way: If you have the chance to get a free Amiga, I'd have a look at the Amiga specific software (especially games and demos) first. Focus on the possibility to transfer to and store files on your Amiga first. If you have some spare money (later) you can use it to upgrade your Amiga to be able to get online or play some more advanced ports.

Ah, sorry, my mistake. I misinterpreted what you was saying (which I believe was actually something to the effect of "horses for courses"?).

With that I cant disagree. If your amiga is a "stock" machine then its probably a better idea to be more concerned/interested in software written with the amiga in mind (ergo will run nicely on a default spec a500/a2000). Once you've exhausted that (a big job given the volume of amiga software available) then perhaps look into upgrading your amiga.

There's all sorts of fun that can be had on an a500/a2000 even when not upgraded.

Oh, one last thing that I omitted before thats probably worth mentioning is that the operating system available to any amiga is also somewhat dictated by the kickstart roms a machine has installed. You cant install os2.x or 3.x on a machine with only kickstart 1.x roms for example, nor can you install os3.x on a machine that has kickstart 2.x roms (not exactly true, there are workarounds, but thats something to worry about once youre more familiar with the amiga (and pretty much requires at least a RAM upgrade beyond 1meg).