Out of all these methods, FMCB and ESR/HDloader is the best method. All can be used though.

Swap Magic

Go buy Swap Magic 3.6 or 3.8 to act as a boot disk. You will also need a set of Magic Switches or a flip top mod for your PS2 to trick the console into thinking that the disc trau is always closed. More info here:

http://ps2iso.net/showthread.php?t=1701

Mod-chip

http://ps2iso.net/showthread.php?t=1699

Another way is to use a Mod Chip and solder it in yourself because I doubt you'll find a shop to do it close to your area, due to most modchip use obviously being illegal. More info? Well you don't need any other than how to obtain one and how you're going to get it installed.

Common solder modchips are:
Crystal Chip
DMS4/3
Matrix Infinity
Magic series

A solderless modchip also exists called the DMS4 Pro EZI, but it is not recommended as it is as hard to install a solder modchip. Except you don't solder.

Memory Card Mod-chip

There is also a new type of memory card modchip called Memor32. Whilst not really a modchip, after flashing a custom firmware to it you can direct boot backups without swapping or anything, just burn and play!

It is generally accepted that this is superseded by ESR. Read on...

Exploit (for older, fat PS2s)

Another way is to use independence card exploit found here:

http://ps2iso.net/showthread.php?t=1755

This way inserts a file on the memory card. A PSX game tries to read the file which triggers an exploit, and the exploit boots up a program called uLaunchElf. One of uLE's functions is a filebrowser that lets you load ELFs (think EXEs, but for a PS2) such as HDLoader, Swap Magic or ESR.

ESR

ESR is a method of directly booting backup games. You launch it from the independence exploit, FMCB (Free Memory Card Boot) or a pressed copy of Swap Magic. This works on both fat and slim PS2s. The main disadvantage of this is that you need to patch all your ISOs to work on it before you burn em. Topics are in the PS2 Apps section http://www.ps2iso.net/index.php?board=59.0
For slim PS2s, there are some cheat devices available that can boot ELFs that will allow you to install FMCB and ESR. Action Replay MAX with Media Player can. (it's the media player bit of AR MAX that's used to boot ELFs) There are others, such as Gameshark, although as to what version, I don't know. For fat PS2s, you can use any other soft mod method that launches elfs, e.g. the independence exploit.

Hard Drive

For a fat PS2, this method involves buying a network adapter and an IDE hard drive. See more information on it at http://ps2iso.net/showthread.php?t=1834 . Not all IDE hard drives work on the PS2, check the linked FAQ.

For a slim PS2, this involves buying something called HDCombo which allows you to use an IDE HDD in a slim PS2. Or you can use a USB HDD, but this has a HUGE variety of sucess, most of the time, load times and compatibility being unacceptable, and other times, working at the same speed as the original DVD with fair compatibility. See the HDD FAQ for more info.

The Games

Now, once that is done, find the game you want (if you can find it) and download it from the file hosting source... ALL the links! Save them all to one folder and un-rar it with WinRAR or whatever Unzipping program you use or one the uploader tells you to use. That will give you a ISO, Image, or MDS(F) file. This is the game, then you get yourself a DVD+/-R blank disk (see the Discs section here) (I prefer Sony DVD+R) and burn it using MagicISO or Nero or Alcohol 120%. Now use whatever boot thingy (I listed 4 at the top) you want and play it.

Discs/Media
--This information is not needed when using the hard drive method.

For more detail, see http://ps2iso.net/showthread.php?t=1753 but the guide is fairly outdated in the specific codes etc. As a rule of thumb, you should only use Verbatim or REAL Taiyo Yuden blank DVDs, which are the highest quality ones. You also need a DVD Burner, and if you pick DVD+R or DVD-R, make sure your burner supports that type. I personally recommend Verbatim, while their best may be inferior to Taiyo Yuden, legitimate Taiyo Yudens are very expensive, and there's no chance of noticing a difference between usage of either brand of disc.

PS2s V4 and below do not support DVD+R by default. See http://ps2iso.net/showthread.php?t=2245 on how to make it work. It's not a complicated process.

Good luck!