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/u/fbjim Virtual Tenho-G for android (may also be available for iPhone) has practice modes to help you get the hang of manual score calculations: there's one mode for scoring full hands, and one just for scoring minipoints (fu).
There's pretty much always some way you can get a winning condition, a yaku - as below, winning off the last tile in the wall (if you draw it), winning off the last discard (as long as you're not furiten at the time), winning off the replacement tile of a kan.
But keeping your hand closed is the best way to get the winning condition - at least 40% of all winning hands are won that way, almost always through riichi. When your hand is closed, you can declare riichi (ready) when you're one tile away, then whether you draw your winning tile it or you claim it when someone else discards it, you get your yaku and win the hand. Even if you don't declare riichi (but you should do so on almost all occasions when you can), you still get a yaku with a fully closed hand when you draw the winning tile yourself (happily, furiten doesn't matter to self-drawn tiles).
The next most common yaku (roughly 30%) is yaku hai: having 3 (or 4) identical tiles of any dragon, or your own wind, or the wind of the round. With this yaku, the hand can be open or closed.
After that, the next most common yaku is all-simples (tan-yao), which can be open or closed: that's about 25% of all winning hands, and consists only of tiles in the range 2-8 of any combination of the three suits.
There are a few ways to get practice in completing yaku. The Flash game is an easier way to get the hang of it, than tenhou, and has a list of all the yaku further down that page. And the Virtual Tenho-G Android app has a practice mode for completing yaku, that you get to by clicking on the wrapped present in the centre of the bottom menu bar in the app. In the top left corner of the screen, it will tell you the name of the yaku to aim for, and it will give you most of the tiles you need to get there.
What version? I only know Riichi and if you want to play solo then Virtual Tenho-G is a good choice. Won't teach you how to play but will help you practice. Even has "hand scoring" practice.
Yaku = "winning pattern". You need at least one winning pattern in order to win a hand. In riichi, unlike other mahjong variants, just having four groups and a pair isn't enough to win a hand. You need that too (except for a couple of odd exceptions) but in addition, you need at least one winning pattern (you can also have more than one in a hand, which will give you a higher score).
The winning patterns are listed under the GameDesign Flash riichi mahjong game, which is one place where you can start to get the hang of this riichi-specific rule.
Another way to get the hang of it is through the Android app Virtual-Tenho-G, which, if you click on the wrapped-present icon at the centre-bottom, has a Practice Mode, where it specifies a particular yaku, and gives you almost all of the tiles you need to get it, and then you just have to complete the hand.
Used to be Janryumon (RIP), and for single player I use Virtual Tenho
Going to try 四人麻雀 FREE to see if it's any good.
I like this for beginners: https://play.google.com/store/apps/details?id=jp.bustercurry.virtualtenho_g&hl=en_GB
t's got a nice practice session to walk you through earning the individual yaku.
It actually sounds like you're making good early progress.
Carry on with the Flash game for now. If you've got a mobile, check out Virtual Tenho-G, which has a practice mode to help you learn the yaku.
Just concentrate on building closed hands at first. And as for when to call riichi: for now, the best rule is: if you can, do, immediately. Later, you'll find exceptions to that; and then exceptions to those exceptions ... but for now, if you can riichi, do so immediately.
Get to learn the yakus one at a time. The Flash game is really good for this. You've learnt what riichi is. In a while, that'll cover about 40% of your wins, so that's a pretty good start.
Then get to know the yaku of a set of three identical value honour tiles (which the Flash game mistakenly calls yaku pai, and which is called in Japanese yaku hai). Value honour tiles, as DZeroX says, are either (a) the wind of the round (East for the first four dealer turns, South for the next four); (b) your own seat wind (which changes every dealer turn); and (c) the dragons. If you have that yaku, then your hand can be open or closed. In a while, you'll have that yaku in quite a proportion of your wins too.
The next yaku to learn is all-simples (tanyao), which means that each and every tile in your hand is in the range 2-8 of any of the suits. That's quite common too.
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Once you've got your head around all of those, it's time for the most complex yaku. But although it's the most complex, it's also very common, so it is important to learn it.
That's pinfu (peace). It's complex because it's not just about the tiles you win on, but how the winning tile fits into your hand.
The names don't matter so much. Each giving winning condition ("yaku") has a Japanese name. It also has several variants of that Japanese name - there's a French transliteration, an English one, and so on. Then there's at least one name that's been formalised by the European Mahjong Association. When you get more into the game, you can start thinking about the names, because that helps you read up on the strategy aspects of the game.
The first important thing is to learn the patterns, and just playing a lot is one good way to start. Another way to learn, which I benefited from, was the practice mode in the free Virtual Tenho-G android game, which gives you almost all the winning tiles you need for a specific winning condition (and it tells you which one, each time), and then you have to play the rest of the hand to complete it. The practice mode is reached after clicking on the wrapped-present icon in the bottom menu bar of the app. The app is mostly, but not entirely translated into English, and has one or two winning conditions that aren't translated, and aren't played in most Western versions of riichi, but don't worry about that.
If you do want to get more into the strategy, then Riichi Book 1 is a free pdf download which is a fantastic guide. And many of us here play on Tenhou.net, which is a riichi site where you can play against humans around the world, at any hour of the year. There are addons for Firefox and Chrome which translate the site into English, and that makes it much easier to get started on the site. There's a Discord chat server where people from here and elsewhere arrange games on tenhou.net, discuss tactics and strategy
As Kyuu says, start with http://gamedesign.jp/flash/mahjong/mahjong_e.html , which contains a list of all the winning patterns (yaku), and will let each of you play on your own and get the hang of the flow of the game. You will need Flash installed, though, to play the game.
If you have an android phone, I also recommend VirtuaTenhou-G, which has a practice mode (click on the wrapped gift box in-app), which gives you practice in completing particular winning patterns. Apparently, you can also use the app to play multiplayer, by networking your phones, but I haven't had chance to test this yet.
You could then each get a free account on tenhou.net , and play each other online in a private lobby. There are browser extensions for Chrome and Firefox that translate pretty much the whole interface to English (you can choose three different degrees of translation - I contributed to, and use, the "thorough" one). If you choose lobby L7447 or L7448, you'll often find an empty room in there, and all four of you enter the room at roughly the same time, you'll get a private game between you. That way, you'll get to play against each other, and the computer will take care of all the shuffling, dealing, scoring, and ensure no rules get accidentally broken.
It's nice to play with real tiles too, but it's much much slower and more cumbersome, and it's easy to break one rule or another without being aware of it.