Symbolism runs on sound
The single most useful thing to know about Chinese symbols: most of them are puns. Same syllable, different character, borrowed meaning.
Bat → good fortune
蝠 (bat) and 福 (fortune) are both fú. So a bat is a wish for luck. Five bats together — 五蝠捧寿 — mean the five blessings: long life, wealth, health, virtue, a peaceful death.
This is why you'll see bats on Qing porcelain and think you're looking at the wrong motif.
Fish → surplus, abundance
鱼 (fish) sounds like 余 (surplus). A whole fish served at New Year, deliberately not finished, means may you have more than you need this year.
年年有余 — surplus every year. The most common New Year greeting image there is.
Bamboo → resilience, integrity
Not a pun — this one is physical. Bamboo bends under snow and springs back; it's hollow, which reads as modesty; it stays green in winter. The scholar's plant.
One of the "three friends of winter" alongside plum and pine.
Vase → peace
瓶 (vase) and 平 (peace) are both píng. Put flowers in one and you get 平安富贵 — peace and wealth. This is why vases turn up in Chinese still-life painting far more often than anyone has flowers to arrange.
Meaning travels through pronunciation, not through the object.