The first and the second column of the table in Wikipedia show endigns for stems on hard and soft consonant. The actual difference is in -ов/-ей and -ом/-ем/-ём:
* Genitive Plural: -ов after a hard consonant, -ей after a soft consonant.
* Instrumental Singular: -ом after a hard consonant, -ем or -ём after a soft consonant, depending on stress. (As far as I know, there is no way to guess the correct stress. You have to remember it.)
All other Cases have pure orthographical variations:
soft consonant + -а = -я
soft consonant + -у = -ю
soft consonant + -ы = -и
soft consonant + -ам = -ям
and so on.
Exceptions:
-хы, -кы, -гы are impossible combinations of sounds in Russian, so we use -хи, -ки, -ги in the Nominative Plural.
грех - грехи, бог - боги.
Ч is always soft, but the orthography does its job here:
Плач (no ь), плача (чя is impossible), плачу (чю is impossible), плачем, о плаче, плачи, плачей, плачам, плачами, о плачах.
Also, -ч + stressed -ём should be written as -чом. Instrumental Singular of ключ is ключом, not ключём.
Ж and ш are hard, but behave as soft:
* -ей in the Genitive Plural
* alternation of -ем/-ом in the Instrumental Singular.
Also, -и in the Nominative Plural, since жы is orthographicaly impossible.
чиж, чижа, чижу, чижом, о чиже, чижи, чижей, чижам, чижами, о чижах
марш, марша, маршу, маршем, о марше, марши, маршей, маршам, маршами, о маршах
So, to summarize the above, the rows for your table:
For -х, -к, -г:
-х, -ха, -ху, -хом, -хе, -хи, -хов, -хам, -хами, -хах
-к, -ка, -ку, -ком, -ке, -ки, -ков, -кам, -ками, -ках
-г, -га, -гу, -гом, -ге, -ги, -гов, -гам, -гами, -гах
For -ч:
-ч, -ча, -чу, -чом/-чем, -че, -чи, -чей, -чам, -чами, -чах
For -ж, -ш:
-ж, -жа, -жу, -жом/-жем, -же, -жи, -жей, -жам, -жами, -жах
-ш, -ша, -шу, -шом/-шем, -ше, -ши, -шей, -шам, -шами, -шах
For hard consonants except for -х, -к, -г, -ж, -ш:
-, -а, -у, -ом, -е, -ы, -ов, -ам, -ами, -ах
For soft consonants except for -ч:
-ь, -я, -ю, -ём/-ем, -е, -и, -ей, -ям, -ями, -ях
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What about -а in the Nominative Plural, it is completely different matter. Quite a number of nouns have -а/-я in the the Nominative Plural instead of -ы/-и.
Бока, глаза, доктора, тополя, кабеля, короба, сервера, слесаря, сектора, договора, крема.
In the colloquial language, you can even hear such weird (and ungrammatical) forms as супа, снайпера, соуса, артикула.
That -а/-я is always stressed.
You have to remember all those words, there is no rule for them.