TEACH US

First, the solver should notice several things:

Next, we can infer a bit about the game logs' format:

Round 1

The text is given away by the first line of the poem, so this round is to help teach you the basic mechanic. Each hand/number pair encodes a letter; the main puzzle is to discover the encoding scheme.

First, notice that the number is always in the range 1-14. Since a Tichu hand consists of 14 cards, this suggests that the numbers correspond to indices within a hand. Indexing gives a card, and we can associate (non-special) cards with the numbers 1-13. The image hand direction will give another bit, together making for 26 possibilities.

Given the information presented, the most natural indexing scheme is order of play. However, it's not clear in which of the four hands we should index.

Knowing the clued phrase, some experimentation yields:

This fully specifies a way to extract a letter. For example, on the top row, we had a 5 and a downwards-facing hand. This means to look at the 5th card played by player 1, which was a 7. A downwards-facing hand on the top row means to add 13, yielding 20. The 20th letter of the alphabet is T. Continuing like that, one recovers the first line of the poem:
T E A C H U S
5 10 1 4 13    6 4
7 1    14 7 4 3
I N L I F E

Round 2

This round was mainly to make sure the solver could execute the mechanic. Doing so yields the clue phrases lucky and skilled.

Round 3

In this round, we must for the first time fill in cards to make the round sensible. This yields the following result:

1(A) A . . .
2233 5566 . . 99TT . . .
5 7 . 8 Q A . . .
88 . . JJ QQ . . .
JP(J) . . .
D(R) . . -
d -
444 . - .
6789T . - 9TJQK . . -
234567 . . -
KK AA . - .
2 K - . .
- - 3

Then, extracting yields the clue phrases wise and good-willed.

Round 4

Again, we fill in the cards:
And so we have

222333 . . .
77799 88844 . . .
2 3 9 A . . - - QQQQ . . .
TTT JJJ . . .
445566 . . .
6 9 . A - KKKK . - .
56789TJ . - .
A

Then, extracting yields the clue phrases husband and wife.

Round 5

Again, we fill in the cards:
And so we have

1(T) T K A . . .
567P(8)9T . . .
23456 . . 23456 . . .
d -
44 99 . . QQ . . .
7788 . . .
KK JJJJ . . .
2233 . . .
T Q K A . . .
D(L) . . .
5 Q A . - .
6789T

And extracting yields the clue phrases grand vows and fulfilled.

Putting it together

With all this, we have the final solution:
Teach us, in life,
to be lucky and skilled,
to be wise and good-willed,
as husband and wife,
to have grand vows fulfilled.