There are mainly two types of playing Go online. One is real time games where you play with another player real time, just like you play face to face, only now you play over the internet. Typically, you log on to the server and there are usually hundreds, if not thousands of players online from all over the world. You can then set a game or join a game and start playing, usually with people around the same strength.
The other type is turn-based Go. These are not real time but players make their moves whenever they have time and log into the server. Players can agree how long to play out a whole game, sometimes one week, sometimes one month or longer depending on the players' schedule. This is like playing a game by mail in the olden days and is very suitable for people who do not have much time to spend online every day or would just like to have a leisurely pace, playing several, sometimes even a dozen or more games at the same time but taking their time to think about the moves at a more relaxed pace.
On the internet, Initially it may be hard to find an opponent because you are still new and do not have a solid rank yet but as you play more, sometimes with bots (i.e. software that plays Go which you can find as their programmers enter them into the system to "train" the software), you will gain a solid rank.
Whichever type you like really depends on your time available and the pace that you like. It can also be a mix of both.
Online real time servers
The KGS Go Server (https://www.gokgs.com)
This server is very popular among North American and European players. One of the popular feature is the ability to create rooms, such national based rooms (there is a Malaysia room which you can join) or based on study groups etc. There are usually many players online and they are generally well mannered.
It sometimes organizes tournaments and also sponsors free lessons. It is a popular place for many teachers, such as Guo Juan which has a room for her students.
You will need to download a client and register for an account. This is really easily done.
Good for Windows, Linux or Mac. My only complain is they are still using Java which I dislike as there is always an annoying message of the Java being out of date etc. But overall, this is a very good and fun place to be, especially if you are a member of a room.
Tygem Go (http://www.tygemgo.com)
Tygem Go is another very popular Korean server especially for Asian players. It is getting very popular and to many people, they like the interface more and the site also looks more modern. The players here are also very strong but it lacks the "room" system as in KGS and as such for those who likes to have a "community" feeling or sometimes just meet friend online and chat, they will prefer KGS.
One of the set back for Tygem is that it is not available for the Mac platform although an iOs app is available which is also very good.
Many Malaysian players also play on Tygem. Do note that from my personal experience, there are many sandbaggers at the low kyu level and as such sometimes an 18k is probably of high kyu strength and unsuspecting beginners may be discouraged. If you are a beginner, KGS is a more friendly server to start with.
WBaduk is a Korean server like Tygem but also like Tygem, the site is in English and has an English client.
Besides playing Go there, the site also has other features such as life and death problems, lectures news and game records. Signing up is easy. Just follow the guide.
However, also as with Tygem, the client is Windows based and Mac users will have problem and unlike Tygem, it does not have an iOs app.
Previously known as IGS (Internet Go Server), it is amongst the oldest Go servers out there. In fact when I started playing Go online more than 10 years ago, I started with IGS. It is now called Pandanet. It is a Japanese Go server and it is similar to KGS although to many players, Pandanet's interface is nicer but in terms of community, KGS is still better. Most of the players here are Japanese.
You download a client which works on Windows, Linux and Macintosh. It also has an Android and iOs app.
If you like a strong sense of community, meeting friends online and sometimes just chatting and playing a casual game, the KGS is a really great place. The people are friendly and sometimes there are also free lectures. Many games especially European games are broadcasted on KGS, sometimes with commentaries. You will see a lot of players online watching games especially during the European Championships.
If you like fierce competition and just wanted to drill your Go and if you are more than just a beginner, then Tygem or WBaduk is the place to go. I would recommend Tygem above WBaduk but you cannot go wrong with both. You will find many players there but beware that there are many sandbaggers there and your experience may not be good if you are a beginner. If you are a beginner and would like to play real time Go online, I still recommend you go to KGS. If you are already a pretty good player having gone past the beginner stage and is aiming at progressing the high kyu and especially the Dan level rank, then Tygem is the place for you to fight it out. Sharks are here.
Pandanet is a nice place to Go too and it has a good interface but the community there is not as strong as in KGS. If you are English speaking, then KGS is a no brainer. But if you would like to play with Japanese players, Pandanet is the place to go.
Online Go Server (https://online-go.com/play)
Although I classify OGS under the turn-based server, you can actually play real time on OGS as well. However, OGS is more well known for its turn-based Go.
The interface is good and you can create your own community. There is no need to download any clients as it is a web-based interface. It is a good place for beginners to start although sometimes finding an opponent is not as easy.
You may also set up or join tournaments as well as create your own group. The TAR UC Weiqi Co-Curricular class has a group at OGS.
Dragon Go Server (http://www.dragongoserver.net)
The Dragon Go Server is the older turn-based Go server. In fact when I started playing Go and wanted to play turn based Go due to my hectic time schedule, I started with DGS. It has many players and they are very pleasant.
The interface may seems a bit out-dated compared to OGS but the players on DGS are really strong and solid. I suggest you give it a try and there are many players on DGS.
against the computer
Ok, sometimes you may not want to play on the internet with real people. Sometimes you don't have internet access but wanted to pass time playing Go. Then this is the software for you.
Computer Go used to be really bad and the only thing they really do is to instill bad habits in you. However, CrazyStone is really good and really strong too. At the highest level, it is of a very strong amateur Dan level. It has now a solid 6 Dan ranking on KGS.
There are a few versions. One for Windows, one for iOs and one for Android. There is no Macintosh version but heck, the iOs version called Champion Go HD which is available at the AppStore is really good in itself. It is of high kyu rank, perhaps even 1 dan at its highest level. If you have nothing to do and has an iPad or an Android device, challenge it and you will have a pretty good game.
The Windows version is really, really strong especially so if you have the right processing power. It also has a very nice interface.